6555:
6088:
6550:{\displaystyle {\Biggl (}{\begin{pmatrix}1&1\\0&0\end{pmatrix}}{\begin{pmatrix}0&0\\1&1\end{pmatrix}}{\Biggr )}^{+}={\begin{pmatrix}1&1\\0&0\end{pmatrix}}^{+}={\begin{pmatrix}{\tfrac {1}{2}}&0\\{\tfrac {1}{2}}&0\end{pmatrix}}\quad \neq \quad {\begin{pmatrix}{\tfrac {1}{4}}&0\\{\tfrac {1}{4}}&0\end{pmatrix}}={\begin{pmatrix}0&{\tfrac {1}{2}}\\0&{\tfrac {1}{2}}\end{pmatrix}}{\begin{pmatrix}{\tfrac {1}{2}}&0\\{\tfrac {1}{2}}&0\end{pmatrix}}={\begin{pmatrix}0&0\\1&1\end{pmatrix}}^{+}{\begin{pmatrix}1&1\\0&0\end{pmatrix}}^{+}}
10094:
9758:
10089:{\displaystyle \left.{\frac {\mathrm {d} }{\mathrm {d} x}}\right|_{x=x_{0}\!\!\!\!\!\!\!}A^{+}=-A^{+}\left({\frac {\mathrm {d} A}{\mathrm {d} x}}\right)A^{+}~+~A^{+}A^{+\top }\left({\frac {\mathrm {d} A^{\top }}{\mathrm {d} x}}\right)\left(I-AA^{+}\right)~+~\left(I-A^{+}A\right)\left({\frac {\mathrm {d} A^{\top }}{\mathrm {d} x}}\right)A^{+\top }A^{+},}
3538:
12604:
14442:
15348:
3364:
9167:
5012:
12418:
16278:, we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the
11433:
6016:
10714:
19093:
11938:
14318:
11325:
17140:
to update the inverse of the correlation matrix, which may need less work. In particular, if the related matrix differs from the original one by only a changed, added or deleted row or column, incremental algorithms exist that exploit the relationship.
4824:
17144:
Similarly, it is possible to update the
Cholesky factor when a row or column is added, without creating the inverse of the correlation matrix explicitly. However, updating the pseudoinverse in the general rank-deficient case is much more complicated.
10969:
12104:
5263:
12404:
8320:
15188:
11588:
11731:
10611:
3830:
9003:
14858:
4830:
13912:, that is, it commutes with its conjugate transpose, then its pseudoinverse can be computed by diagonalizing it, mapping all nonzero eigenvalues to their inverses, and mapping zero eigenvalues to zero. A corollary is that
18950:, using the same four conditions as in our definition above. It turns out that not every continuous linear operator has a continuous linear pseudoinverse in this sense. Those that do are precisely the ones whose range is
4606:
19202:
19306:
19254:
12014:
10793:
8250:
5866:
8381:
7409:
7304:
12795:
12212:
4097:
10862:
10413:
15626:
17998:
17556:
16312:
The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implementation (such as that of
11503:
11842:
11652:
10531:
10344:
3533:{\displaystyle \left(A^{\operatorname {T} }\right)^{+}=\left(A^{+}\right)^{\operatorname {T} },\quad \left({\overline {A}}\right)^{+}={\overline {A^{+}}},\quad \left(A^{*}\right)^{+}=\left(A^{+}\right)^{*}.}
3762:
18716:
16986:
14910:
12687:
11330:
8488:
4240:
18799:
3661:
18803:
A large condition number implies that the problem of finding least-squares solutions to the corresponding system of linear equations is ill-conditioned in the sense that small errors in the entries of
8070:
3934:
2879:
and can be obtained by transposing the matrix and replacing the nonzero values with their multiplicative inverses. That this matrix satisfies the above requirement is directly verified observing that
13068:
7983:
17922:
12599:{\displaystyle {\vec {x}}^{+}={\begin{cases}{\vec {0}}^{\operatorname {T} },&{\text{if }}{\vec {x}}={\vec {0}};\\{\vec {x}}^{*}/({\vec {x}}^{*}{\vec {x}}),&{\text{otherwise}}.\end{cases}}}
5111:
4676:
16715:
16121:
13614:
13291:
2644:
2420:
15762:
15106:
14323:
11142:
11069:
5871:
5122:
4835:
4681:
4392:
4320:
126:
are sometimes used as synonyms for the Moore–Penrose inverse of a matrix, but sometimes applied to other elements of algebraic structures which share some but not all properties expected for an
10621:
19002:
5858:
5117:
1463:
1361:
1076:
970:
11846:
17700:
12305:
16904:
16611:
18589:
17871:
17476:
17313:
15868:
14693:
14648:
14546:
7802:
7757:
7500:
7455:
2133:
1022:
698:
514:
12413:
The pseudoinverse of the null (all zero) vector is the transposed null vector. The pseudoinverse of a non-null vector is the conjugate transposed vector divided by its squared magnitude:
652:
16491:
11237:
18069:
612:
18881:
12897:
9277:
9223:
3356:
466:
18542:
16248:
6079:
734:
15997:
545:
17796:
12932:
8440:
8178:
1255:
10867:
10266:
6869:
5757:
5500:
5415:
5327:
4670:
4531:
2062:
1644:
1585:
16654:
9328:
7864:
7563:
12722:
12019:
10466:
9679:
2830:
20776:
8957:
8841:
8751:
8718:
8579:
18188:
16171:
14497:
14437:{\displaystyle {\begin{aligned}C&={\mathcal {F}}\cdot \Sigma \cdot {\mathcal {F}}^{*},\\C^{+}&={\mathcal {F}}\cdot \Sigma ^{+}\cdot {\mathcal {F}}^{*}.\end{aligned}}}
8255:
17625:
16753:
14307:
10269:
3152:
15155:
11777:
9606:
3689:
2969:
2923:
13875:
13832:
13789:
13714:
13391:
8009:
7166:
7092:
3305:
1162:
427:
371:
345:
315:
10210:
9530:
9400:
7639:
2194:
1946:
14061:
18421:
18253:
13968:
is said to be an EP matrix if it commutes with its pseudoinverse. In such cases (and only in such cases), it is possible to obtain the pseudoinverse as a polynomial in
12985:
12841:
2731:
915:
401:
18660:
18490:
18315:
18034:
17592:
15484:
14243:
6633:
6597:
5622:
5562:
2684:
2460:
16781:) has been argued not to be competitive to the method using the SVD mentioned above, because even for moderately ill-conditioned matrices it takes a long time before
16364:
16274:
14207:
14174:
13475:
13152:
11507:
8094:
7923:
6810:
6769:
6736:
6703:
6670:
5692:
5657:
830:
17108:
17050:
15900:
15684:
15393:
14976:
14942:
13536:
13213:
11202:
11172:
8899:
8522:
8122:
4460:
4426:
3099:
2535:
2311:
1976:
1881:
1851:
1821:
1751:
1674:
1526:
1496:
1394:
1288:
1113:
19120:
18981:
18946:
18915:
18385:
17347:
16839:
16808:
16546:
15815:
13645:
13322:
12959:
12140:
11656:
11229:
10536:
10168:
10141:
9753:
9706:
9635:
9561:
9462:
9431:
8410:
8150:
7195:
7016:
6959:
4124:
3961:
3593:
3229:
3055:
2857:
2566:
2342:
2225:
2092:
2003:
1701:
1192:
580:
219:
57:
18624:
18454:
18113:
15929:
15428:
14605:
14019:
8808:
8660:
7594:
3767:
16392:
slightly may turn this zero into a tiny positive number, thereby affecting the pseudoinverse dramatically as we now have to take the reciprocal of a tiny number.
2757:
17823:
14724:
19150:(as opposed to the involution considered elsewhere in the article); do there exist matrices that fail to have pseudoinverses in this sense? Consider the matrix
19140:
18999:. In this more general setting, a given matrix doesn't always have a pseudoinverse. The necessary and sufficient condition for a pseudoinverse to exist is that
18823:
18339:
18275:
18212:
17746:
17722:
17433:
17413:
17393:
17371:
17132:
17074:
17016:
16777:
16515:
16388:
16340:
16193:
16019:
15784:
15650:
15506:
15448:
15177:
15031:
15004:
14717:
14570:
14275:
14139:
14107:
14083:
13988:
13964:
13932:
13904:
13744:
13669:
13499:
13445:
13421:
13346:
13176:
13122:
13098:
12815:
12631:
12298:
12274:
12250:
10994:
10114:
9726:
9486:
9352:
8981:
8865:
8775:
8627:
8603:
8546:
7888:
7709:
7685:
7661:
7114:
7040:
6985:
6924:
6892:
5584:
5524:
5439:
5354:
4144:
3981:
3562:
3256:
3198:
3024:
2989:
2877:
2777:
2503:
2483:
2277:
2253:
2153:
1901:
1791:
1771:
1721:
885:
858:
794:
770:
274:
250:
184:
84:
20963:
19799:
Liu, Shuangzhe; Trenkler, Götz; Kollo, Tõnu; von Rosen, Dietrich; Baksalary, Oskar Maria (2023). "Professor Heinz
Neudecker and matrix differential calculus".
149:) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra.
3986:
15513:
15343:{\displaystyle A^{+}=\left(A^{*}A\right)^{-1}A^{*}\quad \Leftrightarrow \quad \left(A^{*}A\right)A^{+}=A^{*}\quad \Leftrightarrow \quad R^{*}RA^{+}=A^{*}}
17169:, however, alternative implementations by QR or even the use of an explicit inverse might be preferable, and custom implementations may be unavoidable.
20541:
20148:
19480:
19980:
Söderström, Torsten; Stewart, G. W. (1974). "On the
Numerical Properties of an Iterative Method for Computing the Moore–Penrose Generalized Inverse".
19660:
Maciejewski, Anthony A.; Klein, Charles A. (1985). "Obstacle
Avoidance for Kinematically Redundant Manipulators in Dynamically Varying Environments".
19879:
9162:{\displaystyle A^{+}=\lim _{\delta \searrow 0}\left(A^{*}A+\delta I\right)^{-1}A^{*}=\lim _{\delta \searrow 0}A^{*}\left(AA^{*}+\delta I\right)^{-1}}
19888:
14980:
and their inverses explicitly is often a source of numerical rounding errors and computational cost in practice. An alternative approach using the
4540:
19153:
4149:
19259:
19207:
18738:
12230:
It is also possible to define a pseudoinverse for scalars and vectors. This amounts to treating these as matrices. The pseudoinverse of a scalar
3600:
19731:
Golub, G. H.; Pereyra, V. (April 1973). "The
Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate".
11945:
10719:
8185:
5007:{\displaystyle {\begin{aligned}\left(A^{+}ABB^{*}\right)^{*}&=A^{+}ABB^{*},\\\left(A^{*}ABB^{+}\right)^{*}&=A^{*}ABB^{+}.\end{aligned}}}
3844:
19417:
8325:
7310:
7209:
16495:
which is sometimes referred to as hyper-power sequence. This recursion produces a sequence converging quadratically to the pseudoinverse of
12727:
12145:
10798:
10349:
5018:
16026:
13543:
13220:
3171:
2573:
2349:
20662:
18548:
This result is easily extended to systems with multiple right-hand sides, when the
Euclidean norm is replaced by the Frobenius norm. Let
17929:
17830:
This result is easily extended to systems with multiple right-hand sides, when the
Euclidean norm is replaced by the Frobenius norm. Let
17483:
10280:
Since for invertible matrices the pseudoinverse equals the usual inverse, only examples of non-invertible matrices are considered below.
4324:
4252:
11443:
4249:
The computation of the pseudoinverse is reducible to its construction in the
Hermitian case. This is possible through the equivalences:
11785:
11595:
10474:
10287:
1401:
1299:
17375:
that solves the system may not exist, or if one does exist, it may not be unique. More specifically, a solution exists if and only if
20687:
17137:
11428:{\displaystyle A^{+}={\frac {1}{\sqrt {2}}}\,\mathbf {e} _{1}\left({\frac {\mathbf {e} _{1}+\mathbf {e} _{2}}{\sqrt {2}}}\right)^{*}}
3699:
20909:
18665:
16909:
14871:
12636:
12613:
The pseudoinverse of a squared diagonal matrix is obtained by taking the reciprocal of the nonzero diagonal elements. Formally, if
8447:
6011:{\displaystyle {\begin{aligned}\left(AA^{*}\right)^{+}&=A^{+*}A^{+},\\\left(A^{*}A\right)^{+}&=A^{+}A^{+*}.\end{aligned}}}
20810:
20720:
18835:
In order to solve more general least-squares problems, one can define Moore–Penrose inverses for all continuous linear operators
18387:
with non-unique solutions (such as under-determined systems), the pseudoinverse may be used to construct the solution of minimum
16418:
10268:, etc.). For a complex matrix, the transpose is replaced with the conjugate transpose. For a real-valued symmetric matrix, the
8018:
20682:
20534:
12990:
21055:
20430:
20384:
20361:
19596:
19550:
160:
numbers. Given a rectangular matrix with real or complex entries, its pseudoinverse is unique. It can be computed using the
19449:(1951). "Application of calculus of matrices to method of least squares; with special references to geodetic calculations".
7938:
17883:
10709:{\displaystyle A\,A^{+}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}
3162:. Generalized inverses always exist but are not in general unique. Uniqueness is a consequence of the last two conditions.
19088:{\displaystyle \operatorname {rank} (A)=\operatorname {rank} \left(A^{*}A\right)=\operatorname {rank} \left(AA^{*}\right)}
16659:
1203:
20884:
15694:
15038:
11933:{\displaystyle A^{+}={\begin{pmatrix}{\frac {1}{4}}&{\frac {1}{4}}\\{\frac {1}{4}}&{\frac {1}{4}}\end{pmatrix}}.}
11074:
11001:
4616:
20456:
7808:
7507:
20403:
19343:
17265:
17259:
5767:
3155:
1033:
927:
21060:
20641:
20527:
19783:
18122:
17634:
15185:, may be used. Multiplication by the inverse is then done easily by solving a system with multiple right-hand sides,
16846:
16553:
11320:{\displaystyle A={\sqrt {2}}\left({\frac {\mathbf {e} _{1}+\mathbf {e} _{2}}{\sqrt {2}}}\right)\mathbf {e} _{1}^{*}}
20954:
20672:
19934:
Ben-Israel, Adi; Cohen, Dan (1966). "On
Iterative Computation of Generalized Inverses and Associated Projections".
18553:
17835:
17443:
17277:
15832:
14657:
14612:
14510:
7766:
7721:
7464:
7419:
2097:
986:
662:
478:
13837:
1124:
617:
20599:
7892:
is an orthogonal projection matrix, then its pseudoinverse trivially coincides with the matrix itself, that is,
3841:
The following identity formula can be used to cancel or expand certain subexpressions involving pseudoinverses:
20819:
18039:
7459:
is
Hermitian and idempotent (true if and only if it represents an orthogonal projection), then, for any matrix
587:
18840:
14212:
12846:
9232:
9178:
8869:. The element of this subspace that has the smallest length (that is, is closest to the origin) is the answer
4819:{\displaystyle {\begin{aligned}A^{+}ABB^{*}A^{*}&=BB^{*}A^{*},\\BB^{+}A^{*}AB&=A^{*}AB.\end{aligned}}}
3313:
436:
20914:
19842:
18495:
16320:
The above procedure shows why taking the pseudoinverse is not a continuous operation: if the original matrix
16200:
6025:
707:
20132:
17241:, the LinearAlgebra package of the standard library provides an implementation of the Moore–Penrose inverse
17153:
High-quality implementations of SVD, QR, and back substitution are available in standard libraries, such as
15934:
10964:{\displaystyle A^{+}A\,A^{+}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\0&0\end{pmatrix}}=A^{+}}
523:
20899:
20874:
20796:
20631:
17755:
17320:
17238:
16131:
12902:
8419:
2691:
161:
18089:
12099:{\displaystyle A^{+}={\begin{pmatrix}1&0&0\\0&{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}
8163:
5258:{\displaystyle {\begin{aligned}A^{+}AB&=B(AB)^{+}AB,\\BB^{+}A^{*}&=A^{*}AB(AB)^{+}.\end{aligned}}}
20713:
16251:
10215:
6815:
6636:
5703:
5446:
5361:
5273:
4477:
2008:
1590:
1531:
16616:
12399:{\displaystyle x^{+}={\begin{cases}0,&{\mbox{if }}x=0;\\x^{-1},&{\mbox{otherwise}}.\end{cases}}}
9298:
8315:{\displaystyle \mathbb {K} ^{m}=\operatorname {ran} A\oplus \left(\operatorname {ran} A\right)^{\perp }}
223:
20978:
20958:
20939:
20585:
17269:
16400:
12692:
10418:
9642:
3361:
Pseudoinversion commutes with transposition, complex conjugation, and taking the conjugate transpose:
2782:
138:
8908:
8817:
8727:
8669:
8555:
20879:
20744:
20636:
17157:. Writing one's own implementation of SVD is a major programming project that requires a significant
16137:
14461:
20118:
17597:
16722:
14288:
12449:
12327:
20550:
19542:
19353:
18993:
17200:
15114:
11736:
9576:
3668:
3104:
2928:
2882:
13794:
13751:
13676:
13353:
7992:
7123:
7049:
3267:
410:
354:
328:
298:
20859:
17158:
15182:
10173:
9495:
9365:
9170:
8155:
7599:
2164:
1906:
835:
227:
20495:
20422:
14024:
11583:{\displaystyle A^{+}={\begin{pmatrix}{\frac {1}{2}}&-{\frac {1}{2}}\\0&0\end{pmatrix}}.}
21050:
21018:
20889:
20706:
19706:
18393:
18219:
15109:
12964:
12820:
3101:, is known as a generalized inverse. If the matrix also satisfies the second condition, namely
2697:
894:
380:
134:
18629:
18459:
18284:
18003:
17561:
15453:
11726:{\displaystyle A^{+}={\begin{pmatrix}{\frac {1}{5}}&{\frac {2}{5}}\\0&0\end{pmatrix}}}
10606:{\displaystyle A^{+}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\\0&0\end{pmatrix}}}
6602:
6566:
5591:
5531:
2653:
2429:
20595:
20341:
16349:
16259:
14179:
14146:
13454:
13131:
12987:
rectangular matrix whose diagonal elements are the reciprocal of the original ones, that is,
8127:
8099:
8079:
7895:
6929:
6777:
6741:
6708:
6675:
6642:
5664:
5629:
3825:{\displaystyle \operatorname {ran} \left(A^{+}\right)=\operatorname {ran} \left(A^{*}\right)}
2156:
803:
279:
19776:
Complex-valued matrix derivatives: with applications in signal processing and communications
19534:
17083:
17025:
15875:
15659:
15368:
14951:
14917:
13511:
13188:
11177:
11147:
8874:
8497:
8107:
4435:
4401:
2510:
2286:
1951:
1856:
1826:
1796:
1726:
1649:
1501:
1471:
1369:
1263:
1088:
21024:
20590:
20514:
20157:
19989:
19943:
19740:
19626:
19489:
19098:
18959:
18924:
18893:
18358:
18344:
17211:
function calculates a pseudoinverse using the singular value decomposition provided by the
16817:
16786:
16524:
15793:
15353:
14853:{\displaystyle A^{+}=C^{+}B^{+}=C^{*}\left(CC^{*}\right)^{-1}\left(B^{*}B\right)^{-1}B^{*}}
14502:
13623:
13300:
12937:
12118:
11207:
10146:
10119:
9731:
9684:
9613:
9539:
9440:
9409:
8388:
8135:
7173:
6994:
6937:
6897:
4102:
3939:
3571:
3207:
3065:
3033:
2835:
2544:
2320:
2203:
2070:
1981:
1679:
1170:
558:
197:
62:
35:
18600:
18430:
15905:
15404:
14581:
13995:
12112:
8784:
8636:
7570:
3696:
The kernel and image of the pseudoinverse coincide with those of the conjugate transpose:
8:
20760:
20752:
20569:
19922:
19535:
19530:
19316:
18989:
13503:
13180:
9290:
6901:
2736:
747:
739:
550:
320:
122:
20461:
20161:
19993:
19947:
19744:
19630:
19493:
17805:
20415:
20373:
20257:
20219:
20211:
20173:
20005:
19959:
19905:
19756:
19687:
19348:
19125:
18808:
18324:
18260:
18197:
17731:
17707:
17418:
17398:
17378:
17356:
17166:
17162:
17117:
17059:
17001:
16762:
16500:
16373:
16325:
16178:
16130:
A computationally simple and accurate way to compute the pseudoinverse is by using the
16004:
15769:
15635:
15491:
15433:
15162:
15016:
14989:
14702:
14575:
14555:
14260:
14124:
14092:
14068:
13973:
13949:
13917:
13889:
13729:
13724:
This is a special case of either full column rank or full row rank (treated above). If
13654:
13484:
13430:
13406:
13331:
13161:
13107:
13083:
12800:
12616:
12283:
12259:
12235:
10998:
is neither injective nor surjective, and thus the pseudoinverse cannot be computed via
10979:
10099:
9711:
9471:
9337:
8966:
8850:
8760:
8612:
8588:
8531:
7873:
7694:
7670:
7646:
7099:
7025:
6970:
6909:
6877:
5569:
5509:
5424:
5339:
4129:
3966:
3547:
3241:
3183:
3009:
2974:
2862:
2762:
2488:
2468:
2262:
2238:
2138:
1886:
1776:
1756:
1706:
870:
843:
779:
755:
259:
235:
169:
109:
69:
9289:
In contrast to ordinary matrix inversion, the process of taking pseudoinverses is not
1080:
satisfying all of the following four criteria, known as the Moore–Penrose conditions:
20973:
20944:
20492:
20473:
20426:
20399:
20380:
20357:
20316:
20311:
20294:
20261:
20223:
20177:
20028:
19779:
19642:
19592:
19546:
19147:
15357:
14280:
3261:
113:
19691:
19431:
21006:
20949:
20929:
20605:
20476:
20349:
20306:
20249:
20203:
20165:
20100:
20073:
20047:
19997:
19951:
19897:
19859:
19851:
19804:
19748:
19677:
19669:
19634:
19497:
19426:
19312:
18732:
15398:
14981:
14253:
1293:
20146:
Penrose, Roger (1956). "On best approximate solution of linear matrix equations".
19855:
19840:
Bajo, I. (2021). "Computing Moore–Penrose Inverses with Polynomials in Matrices".
19412:
17112:
for full column rank) is already known, the pseudoinverse for matrices related to
21012:
20968:
20919:
20904:
20864:
20831:
20801:
20646:
19588:
19446:
19338:
16306:
919:
127:
101:
21000:
20994:
20934:
20924:
20849:
20768:
20564:
20337:
20064:
Meyer, Carl D. Jr. (1973). "Generalized inverses and ranks of block matrices".
19808:
19673:
19328:
18388:
18072:
17628:
17435:
is injective. The pseudoinverse solves the "least-squares" problem as follows:
16412:
8961:
and projecting it orthogonally onto the orthogonal complement of the kernel of
157:
146:
93:
24:
20253:
20169:
19502:
19475:
15902:
is normal and, as a consequence, an EP matrix. One can then find a polynomial
13936:
commuting with its transpose implies that it commutes with its pseudoinverse.
9608:
be a real-valued differentiable matrix function with constant rank at a point
21044:
20815:
20729:
20610:
20320:
20295:"Generalized inverses of matrices with entries taken from an arbitrary field"
20032:
19646:
19614:
19526:
19471:
18886:
13909:
4601:{\displaystyle A\in \mathbb {K} ^{m\times n},\ B\in \mathbb {K} ^{n\times p}}
189:
105:
19968:
19197:{\displaystyle A={\begin{bmatrix}1&i\end{bmatrix}}^{\operatorname {T} }}
19122:
denotes the result of applying the involution operation to the transpose of
12843:
rectangular matrix whose only nonzero elements are on the diagonal, meaning
2690:
In the more general case, the pseudoinverse can be expressed leveraging the
152:
The pseudoinverse is defined for all rectangular matrices whose entries are
21029:
20854:
20519:
19301:{\displaystyle \operatorname {rank} \left(A^{\operatorname {T} }A\right)=0}
19249:{\displaystyle \operatorname {rank} \left(AA^{\operatorname {T} }\right)=1}
18996:
10468:, since multiplication by a zero matrix would always produce a zero matrix.
1197:
1117:
need not be the general identity matrix, but it maps all column vectors of
743:
12009:{\displaystyle A={\begin{pmatrix}1&0\\0&1\\0&1\end{pmatrix}},}
10788:{\displaystyle A\,A^{+}A={\begin{pmatrix}1&0\\1&0\end{pmatrix}}=A}
8245:{\displaystyle \mathbb {K} ^{n}=\left(\ker A\right)^{\perp }\oplus \ker A}
7689:
by verifying that the defining properties of the pseudoinverse hold, when
20278:
Hagen, Roland; Roch, Steffen; Silbermann, Bernd (2001). "Section 2.1.2".
20091:
Meyer, Carl D. Jr. (1973). "Generalized inversion of modified matrices".
19408:
18728:
9357:
3062:
A matrix satisfying only the first of the conditions given above, namely
153:
97:
20:
20194:
Planitz, M. (October 1979). "Inconsistent systems of linear equations".
18347:, all of whose infinitude of solutions are given by this last equation.
17020:
has full row or column rank, and the inverse of the correlation matrix (
2229:
can be given a particularly simple algebraic expression. In particular:
20465:
20450:
20446:
20215:
20009:
19963:
19909:
19760:
19333:
18951:
17223:
16283:
14312:
14119:
This is a special case of a normal matrix with eigenvalues 0 and 1. If
8376:{\displaystyle A:\left(\ker A\right)^{\perp }\to \operatorname {ran} A}
7404:{\displaystyle A^{*}\left(I-AA^{+}\right)=\left(I-A^{+}A\right)A^{*}=0}
7299:{\displaystyle A\,\ \left(I-A^{+}A\right)=\left(I-AA^{+}\right)A\ \ =0}
19864:
16403:
exist for calculating the pseudoinverse of block-structured matrices.
12790:{\displaystyle D^{+}={\tilde {D}}^{-1}\oplus \mathbf {0} _{k\times k}}
12207:{\displaystyle A^{+}A={\begin{pmatrix}1&0\\0&1\end{pmatrix}}.}
11234:
Nonetheless, the pseudoinverse can be computed via SVD observing that
10415:
The uniqueness of this pseudoinverse can be seen from the requirement
4092:{\displaystyle A^{+}={}A^{+}{}A^{+*}{}A^{*}{}={}A^{*}{}A^{+*}{}A^{+},}
20869:
20626:
20500:
20481:
19682:
19319:. This abstract definition coincides with the one in linear algebra.
16812:
enters the region of quadratic convergence. However, if started with
14209:, then the pseudoinverse trivially coincides with the matrix itself:
10857:{\displaystyle A^{+}A={\begin{pmatrix}1&0\\0&0\end{pmatrix}}}
10408:{\displaystyle A^{+}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}.}
8903:
we are looking for. It can be found by taking an arbitrary member of
20207:
20104:
20077:
20001:
19955:
19901:
19752:
19638:
15691:
The case of full row rank is treated similarly by using the formula
15621:{\displaystyle A^{*}A\,=\,(QR)^{*}(QR)\,=\,R^{*}Q^{*}QR\,=\,R^{*}R,}
290:
In the following discussion, the following conventions are adopted.
20353:
17993:{\displaystyle \|AX-B\|_{\mathrm {F} }\geq \|AZ-B\|_{\mathrm {F} }}
17551:{\displaystyle \left\|Ax-b\right\|_{2}\geq \left\|Az-b\right\|_{2}}
20509:
18255:. If the latter holds, then the solution is unique if and only if
11498:{\displaystyle A={\begin{pmatrix}1&0\\-1&0\end{pmatrix}},}
20457:
Interactive program & tutorial of Moore–Penrose Pseudoinverse
19820:
19818:
14283:, that is, the singular values are the Fourier coefficients. Let
11837:{\displaystyle A={\begin{pmatrix}1&1\\1&1\end{pmatrix}},}
11647:{\displaystyle A={\begin{pmatrix}1&0\\2&0\end{pmatrix}},}
10526:{\displaystyle A={\begin{pmatrix}1&0\\1&0\end{pmatrix}},}
10339:{\displaystyle A={\begin{pmatrix}0&0\\0&0\end{pmatrix}},}
20698:
20510:
The Moore–Penrose Pseudoinverse. A Tutorial Review of the Theory
18988:
A notion of pseudoinverse exists for matrices over an arbitrary
17215:
function in the base R package. An alternative is to employ the
17203:
provides a calculation of the Moore–Penrose inverse through the
20677:
20667:
17226:
provides a pseudoinverse through the standard package function
17154:
16406:
16314:
16279:
3757:{\displaystyle \ker \left(A^{+}\right)=\ker \left(A^{*}\right)}
3310:
The pseudoinverse of the pseudoinverse is the original matrix:
20777:
Fashion, Faith, and Fantasy in the New Physics of the Universe
20490:
19815:
17800:
is a zero matrix. The solution with minimum Euclidean norm is
1793:). In fact, the above four conditions are fully equivalent to
133:
A common use of the pseudoinverse is to compute a "best fit" (
20048:"Updating Inverse of a Matrix When a Column is Added/Removed"
19308:. So this matrix doesn't have a pseudoinverse in this sense.
18711:{\displaystyle \|Z\|_{\mathrm {F} }\leq \|X\|_{\mathrm {F} }}
18079:
17726:; this provides an infinitude of minimizing solutions unless
17189:
17173:
16981:{\displaystyle A_{0}:=\left(A^{*}A+\delta I\right)^{-1}A^{*}}
14905:{\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}
14063:
can be easily obtained from the characteristic polynomial of
12682:{\displaystyle D={\tilde {D}}\oplus \mathbf {0} _{k\times k}}
8989:
8483:{\displaystyle \left(\operatorname {ran} A\right)^{\perp }.}
4235:{\displaystyle A^{*}={}A^{*}{}A{}A^{+}{}={}A^{+}{}A{}A^{*}.}
20027:(PhD dissertation). Georg-August-Universität zu Göttingen.
18794:{\displaystyle {\mbox{cond}}(A)=\|A\|\left\|A^{+}\right\|.}
18350:
17176:
provides a pseudoinverse calculation through its functions
15363:
The Cholesky decomposition may be computed without forming
12592:
12392:
9764:
3656:{\displaystyle \left(\alpha A\right)^{+}=\alpha ^{-1}A^{+}}
19146:: Consider the field of complex numbers equipped with the
17631:. This weak inequality holds with equality if and only if
8065:{\displaystyle A^{+}:\mathbb {K} ^{m}\to \mathbb {K} ^{n}}
3929:{\displaystyle A={}A{}A^{*}{}A^{+*}{}={}A^{+*}{}A^{*}{}A.}
19798:
7020:(which equals the orthogonal complement of the kernel of
19707:"On continuity of the Moore–Penrose and Drazin inverses"
19559:
19525:
18827:
can lead to huge errors in the entries of the solution.
14087:
or, more generally, from any annihilating polynomial of
13063:{\displaystyle A_{ii}\neq 0\implies A_{ii}^{+}=1/A_{ii}}
7203:
The last two properties imply the following identities:
20025:
Worterkennung mit einem künstlichen neuronalen Netzwerk
16344:
has a singular value 0 (a diagonal entry of the matrix
20471:
19169:
18743:
12380:
12339:
12170:
12041:
11960:
11868:
11800:
11678:
11610:
11529:
11458:
10903:
10823:
10748:
10647:
10558:
10489:
10371:
10302:
7978:{\displaystyle A:\mathbb {K} ^{n}\to \mathbb {K} ^{m}}
6510:
6467:
6434:
6413:
6409:
6385:
6364:
6355:
6323:
6302:
6298:
6264:
6243:
6239:
6194:
6140:
6104:
5863:
The fourth sufficient condition yields the equalities
3107:
3068:
2971:, which are the projections onto image and support of
19615:"Note on the Generalized Inverse of a Matrix Product"
19262:
19210:
19156:
19128:
19101:
19005:
18962:
18927:
18896:
18843:
18811:
18741:
18668:
18632:
18603:
18556:
18498:
18462:
18433:
18396:
18361:
18327:
18287:
18263:
18222:
18200:
18125:
18092:
18042:
18006:
17932:
17917:{\displaystyle \forall X\in \mathbb {K} ^{n\times p}}
17886:
17838:
17808:
17758:
17734:
17710:
17637:
17600:
17564:
17486:
17446:
17421:
17401:
17381:
17359:
17323:
17280:
17120:
17086:
17062:
17028:
17004:
16912:
16849:
16820:
16789:
16765:
16725:
16662:
16619:
16556:
16527:
16503:
16421:
16376:
16352:
16328:
16262:
16203:
16181:
16140:
16125:
16029:
16007:
15937:
15908:
15878:
15835:
15796:
15772:
15697:
15662:
15638:
15516:
15494:
15456:
15436:
15407:
15371:
15191:
15165:
15117:
15041:
15019:
14992:
14954:
14920:
14874:
14727:
14705:
14660:
14615:
14584:
14558:
14513:
14464:
14321:
14291:
14263:
14215:
14182:
14149:
14127:
14095:
14071:
14027:
13998:
13976:
13952:
13920:
13892:
13840:
13797:
13754:
13732:
13679:
13657:
13626:
13546:
13540:
is invertible. In this case, an explicit formula is:
13514:
13487:
13457:
13433:
13409:
13356:
13334:
13303:
13223:
13217:
is invertible. In this case, an explicit formula is:
13191:
13164:
13134:
13110:
13086:
12993:
12967:
12940:
12905:
12849:
12823:
12803:
12730:
12695:
12639:
12619:
12421:
12308:
12286:
12262:
12238:
12148:
12121:
12022:
11948:
11849:
11788:
11739:
11659:
11598:
11510:
11446:
11333:
11240:
11210:
11180:
11150:
11077:
11004:
10982:
10870:
10801:
10722:
10624:
10539:
10477:
10421:
10352:
10290:
10218:
10176:
10149:
10122:
10102:
9761:
9734:
9714:
9687:
9645:
9616:
9579:
9542:
9498:
9474:
9443:
9412:
9368:
9340:
9301:
9235:
9181:
9006:
8969:
8911:
8877:
8853:
8820:
8787:
8763:
8730:
8672:
8639:
8615:
8591:
8558:
8534:
8500:
8450:
8422:
8391:
8328:
8258:
8188:
8166:
8138:
8110:
8082:
8021:
7995:
7941:
7898:
7876:
7811:
7769:
7724:
7697:
7673:
7649:
7602:
7573:
7510:
7467:
7422:
7313:
7212:
7176:
7126:
7102:
7052:
7028:
6997:
6973:
6940:
6912:
6880:
6818:
6780:
6744:
6711:
6678:
6645:
6605:
6569:
6091:
6028:
5869:
5770:
5706:
5667:
5632:
5594:
5572:
5534:
5512:
5449:
5427:
5364:
5342:
5276:
5120:
5106:{\displaystyle A^{+}ABB^{*}A^{*}ABB^{+}=BB^{*}A^{*}A}
5021:
4833:
4679:
4619:
4543:
4480:
4438:
4404:
4327:
4255:
4152:
4132:
4105:
3989:
3969:
3942:
3847:
3770:
3702:
3671:
3603:
3574:
3550:
3367:
3316:
3270:
3244:
3210:
3186:
3036:
3012:
2977:
2931:
2885:
2865:
2838:
2785:
2765:
2739:
2700:
2656:
2576:
2547:
2513:
2491:
2471:
2432:
2352:
2323:
2289:
2265:
2241:
2206:
2167:
2141:
2100:
2073:
2011:
1984:
1954:
1909:
1889:
1859:
1829:
1799:
1779:
1759:
1729:
1709:
1682:
1652:
1593:
1534:
1504:
1474:
1404:
1372:
1302:
1266:
1206:
1173:
1127:
1091:
1036:
989:
930:
897:
873:
846:
806:
782:
758:
710:
665:
620:
590:
561:
526:
481:
439:
413:
383:
357:
331:
301:
262:
238:
200:
192:(for example, a Hermitian matrix), the pseudoinverse
172:
72:
38:
20417:
Generalized Inverse of Matrices and its Applications
19778:. New York: Cambridge university press. p. 52.
16710:{\displaystyle 0<\alpha <2/\sigma _{1}^{2}(A)}
16411:
Another method for computing the pseudoinverse (cf.
16116:{\displaystyle A^{+}=p(A^{*}A)A^{*}=A^{*}p(AA^{*}).}
15764:
and using a similar argument, swapping the roles of
13609:{\displaystyle A^{+}=A^{*}\left(AA^{*}\right)^{-1}.}
13286:{\displaystyle A^{+}=\left(A^{*}A\right)^{-1}A^{*}.}
2639:{\displaystyle A^{+}=A^{*}\left(AA^{*}\right)^{-1}.}
2415:{\displaystyle A^{+}=\left(A^{*}A\right)^{-1}A^{*}.}
20277:
19882:; Boullion, T. L. (1972). "The Pseudoinverse of an
19541:(3rd ed.). Baltimore: Johns Hopkins. pp.
19413:"On the reciprocal of the general algebraic matrix"
15757:{\displaystyle A^{+}=A^{*}\left(AA^{*}\right)^{-1}}
15101:{\displaystyle A^{+}=\left(A^{*}A\right)^{-1}A^{*}}
14279:, the singular value decomposition is given by the
11137:{\displaystyle A^{+}=A^{*}\left(AA^{*}\right)^{-1}}
11064:{\displaystyle A^{+}=\left(A^{*}A\right)^{-1}A^{*}}
10143:and derivatives on the right side are evaluated at
8444:is the inverse of this isomorphism, and is zero on
4387:{\displaystyle A^{+}=A^{*}\left(AA^{*}\right)^{+},}
4315:{\displaystyle A^{+}=\left(A^{*}A\right)^{+}A^{*},}
20414:
20372:
20336:
20240:James, M. (June 1978). "The generalised inverse".
20149:Proceedings of the Cambridge Philosophical Society
19824:
19481:Proceedings of the Cambridge Philosophical Society
19374:
19300:
19248:
19196:
19134:
19114:
19087:
18975:
18940:
18909:
18875:
18817:
18793:
18710:
18654:
18618:
18583:
18536:
18484:
18448:
18415:
18379:
18333:
18309:
18269:
18247:
18206:
18182:
18107:
18063:
18028:
17992:
17916:
17865:
17817:
17790:
17740:
17716:
17694:
17619:
17586:
17550:
17470:
17427:
17407:
17387:
17365:
17341:
17307:
17126:
17102:
17068:
17044:
17010:
16980:
16898:
16833:
16802:
16771:
16747:
16709:
16648:
16605:
16540:
16509:
16485:
16382:
16358:
16334:
16268:
16242:
16187:
16165:
16115:
16013:
15991:
15923:
15894:
15862:
15809:
15778:
15756:
15678:
15644:
15620:
15500:
15478:
15442:
15422:
15387:
15342:
15171:
15149:
15100:
15025:
14998:
14970:
14936:
14904:
14852:
14711:
14687:
14642:
14599:
14564:
14540:
14491:
14436:
14301:
14269:
14237:
14201:
14168:
14133:
14101:
14077:
14055:
14013:
13982:
13958:
13926:
13898:
13869:
13826:
13783:
13738:
13708:
13663:
13639:
13608:
13530:
13493:
13469:
13439:
13415:
13385:
13340:
13316:
13285:
13207:
13170:
13146:
13116:
13092:
13062:
12979:
12953:
12926:
12891:
12835:
12809:
12789:
12716:
12681:
12625:
12598:
12398:
12292:
12268:
12244:
12206:
12134:
12098:
12008:
11932:
11836:
11771:
11725:
11646:
11582:
11497:
11427:
11319:
11223:
11196:
11166:
11136:
11063:
10988:
10963:
10856:
10787:
10708:
10605:
10525:
10460:
10407:
10338:
10260:
10204:
10162:
10135:
10108:
10088:
9747:
9720:
9700:
9673:
9629:
9600:
9555:
9524:
9480:
9456:
9425:
9394:
9346:
9322:
9271:
9217:
9161:
8975:
8951:
8893:
8859:
8835:
8802:
8769:
8745:
8712:
8654:
8621:
8597:
8573:
8540:
8516:
8482:
8434:
8404:
8375:
8314:
8244:
8172:
8144:
8116:
8088:
8064:
8003:
7977:
7917:
7882:
7858:
7796:
7751:
7703:
7679:
7655:
7633:
7588:
7557:
7494:
7449:
7403:
7298:
7189:
7160:
7108:
7086:
7034:
7010:
6979:
6953:
6918:
6886:
6863:
6804:
6763:
6730:
6697:
6664:
6627:
6591:
6549:
6073:
6010:
5852:
5751:
5686:
5651:
5616:
5578:
5556:
5518:
5494:
5433:
5409:
5348:
5321:
5257:
5105:
5006:
4818:
4664:
4600:
4525:
4454:
4420:
4386:
4314:
4234:
4138:
4118:
4091:
3975:
3955:
3928:
3824:
3756:
3683:
3655:
3587:
3556:
3532:
3350:
3299:
3250:
3223:
3192:
3146:
3093:
3049:
3018:
2983:
2963:
2917:
2871:
2851:
2824:
2771:
2751:
2725:
2678:
2638:
2560:
2529:
2497:
2477:
2454:
2414:
2336:
2305:
2271:
2247:
2219:
2188:
2147:
2127:
2086:
2056:
1997:
1970:
1940:
1895:
1875:
1845:
1815:
1785:
1765:
1745:
1715:
1695:
1668:
1638:
1579:
1520:
1490:
1457:
1388:
1355:
1282:
1249:
1186:
1156:
1107:
1070:
1016:
964:
909:
879:
852:
824:
788:
764:
728:
692:
646:
606:
574:
539:
508:
460:
421:
395:
365:
339:
309:
268:
244:
213:
178:
78:
51:
20413:Rao, C. Radhakrishna; Mitra, Sujit Kumar (1971).
19979:
19878:
15999:. In this case one has that the pseudoinverse of
14114:
9812:
9811:
9810:
9809:
9808:
9807:
9806:
6174:
6094:
5853:{\displaystyle (A^{+}A)(BB^{+})=(BB^{+})(A^{+}A)}
1458:{\displaystyle \left(A^{+}A\right)^{*}=\;A^{+}A.}
1356:{\displaystyle \left(AA^{+}\right)^{*}=\;AA^{+}.}
1071:{\displaystyle A^{+}\in \mathbb {K} ^{n\times m}}
965:{\displaystyle I_{n}\in \mathbb {K} ^{n\times n}}
112:had introduced the concept of a pseudoinverse of
92:, is the most widely known generalization of the
16:Most widely known generalized inverse of a matrix
21042:
19889:Proceedings of the American Mathematical Society
19659:
18343:does not have full column rank, then we have an
15822:
14471:
9708:may be calculated in terms of the derivative of
9092:
9021:
3170:Proofs for the properties below can be found at
2257:has linearly independent columns (equivalently,
2168:
19933:
17695:{\displaystyle x=A^{+}b+\left(I-A^{+}A\right)w}
16843:already close to the Moore–Penrose inverse and
13073:
7168:is the orthogonal projector onto the kernel of
7094:is the orthogonal projector onto the kernel of
20396:Advanced Robotics: Redundancy and Optimization
20375:Generalized Inverses of Linear Transformations
19315:, a Moore–Penrose inverse may be defined on a
17245:implemented via singular-value decomposition.
16899:{\displaystyle A_{0}A=\left(A_{0}A\right)^{*}}
16606:{\displaystyle A_{0}A=\left(A_{0}A\right)^{*}}
13719:
6989:is the orthogonal projector onto the range of
4244:
141:that lacks an exact solution (see below under
20714:
20535:
20370:
20346:Generalized inverses: Theory and applications
19872:
19698:
19582:
19565:
19466:
19464:
19439:
19418:Bulletin of the American Mathematical Society
19380:
18584:{\displaystyle B\in \mathbb {K} ^{m\times p}}
17866:{\displaystyle B\in \mathbb {K} ^{m\times p}}
17471:{\displaystyle \forall x\in \mathbb {K} ^{n}}
17308:{\displaystyle A\in \mathbb {K} ^{m\times n}}
16991:
15863:{\displaystyle A\in \mathbb {K} ^{m\times n}}
14688:{\displaystyle C\in \mathbb {K} ^{r\times n}}
14643:{\displaystyle B\in \mathbb {K} ^{m\times r}}
14541:{\displaystyle A\in \mathbb {K} ^{m\times n}}
14143:is an orthogonal projection matrix, that is,
7797:{\displaystyle B\in \mathbb {K} ^{n\times m}}
7752:{\displaystyle A\in \mathbb {K} ^{n\times n}}
7495:{\displaystyle B\in \mathbb {K} ^{m\times n}}
7450:{\displaystyle A\in \mathbb {K} ^{n\times n}}
3264:, its pseudoinverse is its inverse. That is,
2485:has linearly independent rows (equivalently,
2128:{\displaystyle A\in \mathbb {K} ^{m\times n}}
1017:{\displaystyle A\in \mathbb {K} ^{m\times n}}
693:{\displaystyle A\in \mathbb {K} ^{m\times n}}
509:{\displaystyle A\in \mathbb {K} ^{m\times n}}
20549:
19730:
18767:
18761:
18697:
18690:
18676:
18669:
18525:
18518:
18506:
18499:
18404:
18397:
18050:
18043:
17979:
17963:
17949:
17933:
17608:
17601:
16407:The iterative method of Ben-Israel and Cohen
14899:
14883:
13396:
8943:
8928:
8704:
8689:
7761:is Hermitian and idempotent, for any matrix
5268:The following are sufficient conditions for
4467:
2183:
2171:
774:(the space spanned by the column vectors of
647:{\displaystyle A^{*}=A^{\operatorname {T} }}
20890:Penrose interpretation of quantum mechanics
20421:. New York: John Wiley & Sons. p.
20139:
20039:
16486:{\displaystyle A_{i+1}=2A_{i}-A_{i}AA_{i},}
8988:This description is closely related to the
7716:From the last property it follows that, if
5698:The following is a necessary condition for
3172:b:Topics in Abstract Algebra/Linear algebra
2999:
2779:. The pseudoinverse can then be written as
20721:
20707:
20542:
20528:
20371:Campbell, S. L.; Meyer, C. D. Jr. (1991).
20189:
20187:
19773:
19662:International Journal of Robotics Research
19461:
19445:
18080:Obtaining all solutions of a linear system
17219:function available in the pracma package.
13017:
13013:
8383:is then an isomorphism. This implies that
4535:does not hold in general. Rather, suppose
3542:The pseudoinverse of a scalar multiple of
1773:(equivalently, the span of the columns of
1528:are idempotent operators, as follows from
1438:
1336:
1233:
1147:
20310:
19863:
19704:
19681:
19521:
19519:
19517:
19515:
19513:
19501:
19430:
18565:
18319:is a zero matrix. If solutions exist but
18117:has any solutions, they are all given by
18064:{\displaystyle \|\cdot \|_{\mathrm {F} }}
17898:
17847:
17458:
17289:
15844:
15601:
15597:
15570:
15566:
15534:
15530:
14895:
14887:
14876:
14669:
14624:
14522:
12920:
11359:
10884:
10726:
10628:
8823:
8733:
8561:
8261:
8191:
8052:
8037:
7997:
7965:
7950:
7928:
7778:
7733:
7476:
7431:
7216:
4582:
4552:
2109:
1052:
998:
946:
674:
607:{\displaystyle \mathbb {K} =\mathbb {R} }
600:
592:
549:and the Hermitian transpose (also called
490:
442:
415:
359:
333:
303:
20412:
20393:
20348:(2nd ed.). New York, NY: Springer.
20273:
20271:
20235:
20233:
19916:
19839:
19612:
19578:
19576:
19574:
19401:
19392:
19386:
18876:{\displaystyle A:H_{1}\rightarrow H_{2}}
18351:Minimum norm solution to a linear system
12892:{\displaystyle A_{ij}=\delta _{ij}a_{i}}
9272:{\displaystyle \left(A^{*}A\right)^{-1}}
9218:{\displaystyle \left(AA^{*}\right)^{-1}}
8990:minimum-norm solution to a linear system
7567:This can be proven by defining matrices
3351:{\displaystyle \left(A^{+}\right)^{+}=A}
3028:there is one and only one pseudoinverse
461:{\displaystyle \mathbb {K} ^{m\times n}}
20811:The Large, the Small and the Human Mind
20673:Basic Linear Algebra Subprograms (BLAS)
20515:Online Moore–Penrose Inverse calculator
20193:
20184:
20145:
20046:Emtiyaz, Mohammad (February 27, 2008).
20045:
19835:
19833:
19583:Stoer, Josef; Bulirsch, Roland (2002).
19470:
18537:{\displaystyle \|z\|_{2}\leq \|x\|_{2}}
17253:
16757:denoting the largest singular value of
16243:{\displaystyle A^{+}=V\Sigma ^{+}U^{*}}
16173:is the singular value decomposition of
15397:explicitly, by alternatively using the
14313:Discrete Fourier Transform (DFT) matrix
13102:is identical to the number of columns,
8074:can be decomposed as follows. We write
6074:{\displaystyle (AB)^{+}\neq B^{+}A^{+}}
5528:has linearly independent columns (then
1703:(equivalently, the span of the rows of
729:{\displaystyle \operatorname {ran} (A)}
145:). Another use is to find the minimum (
21043:
19510:
15992:{\displaystyle (A^{*}A)^{+}=p(A^{*}A)}
11231:is neither a left nor a right inverse.
7933:If we view the matrix as a linear map
7414:Another property is the following: if
540:{\displaystyle A^{\operatorname {T} }}
20702:
20523:
20491:
20472:
20292:
20268:
20239:
20230:
20090:
20063:
20022:
20016:
19571:
19407:
17791:{\displaystyle \left(I-A^{+}A\right)}
17317:, given a system of linear equations
17148:
16519:if it is started with an appropriate
14451:
14247:
12927:{\displaystyle a_{i}\in \mathbb {K} }
8812:. This will be an affine subspace of
8435:{\displaystyle \operatorname {ran} A}
4610:. Then the following are equivalent:
2759:and diagonal nonnegative real matrix
254:and acts as a traditional inverse of
20910:Penrose–Hawking singularity theorems
20119:"R: Generalized Inverse of a Matrix"
19830:
19476:"A generalized inverse for matrices"
19142:. When it does exist, it is unique.
18279:has full column rank, in which case
17750:has full column rank, in which case
17161:. In special circumstances, such as
13425:is identical to the number of rows,
12608:
8182:for the image of a map. Notice that
8173:{\displaystyle \operatorname {ran} }
5588:has linearly independent rows (then
1250:{\displaystyle A^{+}AA^{+}=\;A^{+}.}
375:, respectively. The vector space of
323:of real or complex numbers, denoted
96:. It was independently described by
20299:Linear Algebra and Its Applications
19923:Linear Systems & Pseudo-Inverse
18722:
18216:. Solution(s) exist if and only if
17196:that uses a least-squares solver.
16988:, convergence is fast (quadratic).
11204:are both singular, and furthermore
10261:{\displaystyle A^{+}:=A^{+}(x_{0})}
6864:{\displaystyle A^{+}P=QA^{+}=A^{+}}
5752:{\displaystyle (AB)^{+}=B^{+}A^{+}}
5495:{\displaystyle BB^{*}=BB^{+}=I_{n}}
5410:{\displaystyle A^{*}A=A^{+}A=I_{n}}
5322:{\displaystyle (AB)^{+}=B^{+}A^{+}}
4665:{\displaystyle (AB)^{+}=B^{+}A^{+}}
4526:{\displaystyle (AB)^{+}=B^{+}A^{+}}
3165:
3004:As discussed above, for any matrix
2422:This particular pseudoinverse is a
2057:{\displaystyle (A^{+}A)A^{+}=A^{+}}
1853:being such orthogonal projections:
1639:{\displaystyle (A^{+}A)^{2}=A^{+}A}
1580:{\displaystyle (AA^{+})^{2}=AA^{+}}
13:
20280:C*-algebras and Numerical Analysis
19982:SIAM Journal on Numerical Analysis
19936:SIAM Journal on Numerical Analysis
19733:SIAM Journal on Numerical Analysis
19587:(3rd ed.). Berlin, New York:
19585:Introduction to Numerical Analysis
19344:Linear least squares (mathematics)
19279:
19230:
19189:
18830:
18702:
18681:
18055:
17984:
17954:
17887:
17447:
17260:Linear least squares (mathematics)
16649:{\displaystyle A_{0}=\alpha A^{*}}
16353:
16263:
16221:
16150:
16126:Singular value decomposition (SVD)
14416:
14401:
14392:
14355:
14346:
14338:
14294:
13879:
12633:is a squared diagonal matrix with
12466:
10068:
10046:
10038:
10029:
9940:
9932:
9923:
9910:
9859:
9849:
9775:
9769:
9323:{\displaystyle \left(A_{n}\right)}
8995:
7859:{\displaystyle (AB)^{+}A=(AB)^{+}}
7558:{\displaystyle A(BA)^{+}=(BA)^{+}}
3416:
3378:
2694:. Any matrix can be decomposed as
639:
532:
14:
21072:
20728:
20462:Moore–Penrose generalized inverse
20440:
19613:Greville, T. N. E. (1966-10-01).
19451:Trans. Roy. Inst. Tech. Stockholm
17138:Sherman–Morrison–Woodbury formula
16395:
12717:{\displaystyle {\tilde {D}}>0}
10461:{\displaystyle A^{+}=A^{+}AA^{+}}
9674:{\displaystyle x\mapsto A^{+}(x)}
8722:, that is, find those vectors in
2825:{\displaystyle A^{+}=VD^{+}U^{*}}
142:
20955:Orchestrated objective reduction
20828:White Mars or, The Mind Set Free
17136:can be computed by applying the
15035:is of full column rank, so that
14863:
12771:
12663:
12220:
11397:
11382:
11362:
11302:
11278:
11263:
8952:{\displaystyle A^{-1}(\{p(b)\})}
8836:{\displaystyle \mathbb {K} ^{n}}
8746:{\displaystyle \mathbb {K} ^{n}}
8713:{\displaystyle A^{-1}(\{p(b)\})}
8574:{\displaystyle \mathbb {K} ^{m}}
20293:Pearl, Martin H. (1968-10-01).
20286:
20125:
20111:
20084:
20057:
19973:
19927:
19792:
19767:
19724:
19432:10.1090/S0002-9904-1920-03322-7
18183:{\displaystyle x=A^{+}b+\leftw}
17415:, and is unique if and only if
17248:
16290:, the tolerance is taken to be
16166:{\displaystyle A=U\Sigma V^{*}}
15303:
15299:
15252:
15248:
14492:{\displaystyle r\leq \min(m,n)}
14446:
9435:. However, if all the matrices
8607:orthogonally onto the range of
6639:, that is, they are Hermitian (
6637:orthogonal projection operators
6292:
6288:
6020:Here is a counterexample where
3473:
3424:
3202:has real entries, then so does
19825:Ben-Israel & Greville 2003
19653:
19606:
19375:Ben-Israel & Greville 2003
19365:
19018:
19012:
18860:
18784:
18771:
18755:
18749:
18727:Using the pseudoinverse and a
17620:{\displaystyle \|\cdot \|_{2}}
17538:
17521:
17506:
17489:
17188:uses the SVD-based algorithm.
16748:{\displaystyle \sigma _{1}(A)}
16742:
16736:
16704:
16698:
16107:
16091:
16062:
16046:
15986:
15970:
15955:
15938:
15918:
15912:
15563:
15554:
15545:
15535:
15300:
15249:
14486:
14474:
14302:{\displaystyle {\mathcal {F}}}
14115:Orthogonal projection matrices
14050:
14044:
14008:
14002:
13939:
13014:
12751:
12702:
12652:
12573:
12567:
12549:
12539:
12522:
12502:
12487:
12459:
12429:
12278:is zero and the reciprocal of
10255:
10242:
10199:
10186:
9668:
9662:
9649:
9595:
9589:
9583:
9513:
9499:
9383:
9369:
9358:maximum norm or Frobenius norm
9173:). These limits exist even if
9099:
9028:
9000:The pseudoinverse are limits:
8946:
8940:
8934:
8925:
8797:
8791:
8707:
8701:
8695:
8686:
8649:
8643:
8361:
8047:
7960:
7847:
7837:
7822:
7812:
7665:is indeed a pseudoinverse for
7622:
7612:
7546:
7536:
7524:
7514:
7504:the following equation holds:
6039:
6029:
5847:
5831:
5828:
5812:
5806:
5790:
5787:
5771:
5717:
5707:
5358:has orthonormal columns (then
5287:
5277:
5239:
5229:
5161:
5151:
4630:
4620:
4491:
4481:
3566:is the reciprocal multiple of
3147:{\textstyle A^{+}AA^{+}=A^{+}}
2028:
2012:
1926:
1910:
1611:
1594:
1552:
1535:
819:
813:
723:
717:
1:
20915:Riemannian Penrose inequality
20330:
19856:10.1080/00029890.2021.1886840
19843:American Mathematical Monthly
18662:is a solution, and satisfies
18492:is a solution, and satisfies
17264:The pseudoinverse provides a
15823:Using polynomials in matrices
15150:{\displaystyle A^{*}A=R^{*}R}
11772:{\displaystyle 5=1^{2}+2^{2}}
9601:{\displaystyle x\mapsto A(x)}
9568:
9284:
7713:is Hermitian and idempotent.
6559:
3836:
3684:{\displaystyle \alpha \neq 0}
2994:
2964:{\displaystyle A^{+}A=VV^{*}}
2918:{\displaystyle AA^{+}=UU^{*}}
1978:projecting onto the image of
1883:projecting onto the image of
978:
21056:Singular value decomposition
20822:and Stephen Hawking) (1997)
20797:The Nature of Space and Time
20394:Nakamura, Yoshihiko (1991).
20312:10.1016/0024-3795(68)90028-1
19705:Rakočević, Vladimir (1997).
17239:Julia (programming language)
16132:singular value decomposition
13870:{\displaystyle A^{+}=A^{*}.}
13827:{\displaystyle AA^{*}=I_{m}}
13784:{\displaystyle A^{*}A=I_{n}}
13709:{\displaystyle AA^{+}=I_{m}}
13386:{\displaystyle A^{+}A=I_{n}}
13074:Linearly independent columns
11733:. The denominators are here
8004:{\displaystyle \mathbb {K} }
7161:{\displaystyle I-P=I-AA^{+}}
7087:{\displaystyle I-Q=I-A^{+}A}
3465:
3435:
3300:{\displaystyle A^{+}=A^{-1}}
2692:singular value decomposition
1157:{\displaystyle AA^{+}A=\;A.}
422:{\displaystyle \mathbb {K} }
366:{\displaystyle \mathbb {C} }
340:{\displaystyle \mathbb {R} }
310:{\displaystyle \mathbb {K} }
164:. In the special case where
162:singular value decomposition
137:) approximate solution to a
7:
19322:
18992:equipped with an arbitrary
18626:is satisfiable, the matrix
18456:is satisfiable, the vector
17224:Octave programming language
16252:rectangular diagonal matrix
13720:Orthonormal columns or rows
10275:
10270:Magnus-Neudecker derivative
10205:{\displaystyle A:=A(x_{0})}
9525:{\displaystyle (A_{n})^{+}}
9395:{\displaystyle (A_{n})^{+}}
7634:{\displaystyle D=A(BA)^{+}}
5443:has orthonormal rows (then
4245:Reduction to Hermitian case
3936:Equivalently, substituting
2189:{\displaystyle \min\{m,n\}}
1941:{\displaystyle (AA^{+})A=A}
1753:projects onto the image of
1676:projects onto the image of
518:, the transpose is denoted
285:
10:
21077:
20979:Conformal cyclic cosmology
20940:Penrose graphical notation
20586:System of linear equations
19809:10.1007/s00362-023-01499-w
19674:10.1177/027836498500400308
17270:system of linear equations
17257:
16992:Updating the pseudoinverse
15654:is the Cholesky factor of
15510:is upper triangular. Then
14056:{\displaystyle A^{+}=p(A)}
13326:is then a left inverse of
12408:
12225:
8845:parallel to the kernel of
139:system of linear equations
20987:
20880:Weyl curvature hypothesis
20842:
20787:
20736:
20655:
20637:Cache-oblivious algorithm
20619:
20578:
20557:
20254:10.1017/S0025557200086460
20170:10.1017/S0305004100030929
19566:Campbell & Meyer 1991
19503:10.1017/S0305004100030401
19381:Campbell & Meyer 1991
18416:{\displaystyle \|x\|_{2}}
18248:{\displaystyle AA^{+}b=b}
15450:has orthonormal columns,
13748:has orthonormal columns (
13397:Linearly independent rows
12980:{\displaystyle n\times m}
12836:{\displaystyle m\times n}
4468:Pseudoinverse of products
2726:{\displaystyle A=UDV^{*}}
910:{\displaystyle n\times n}
865:For any positive integer
396:{\displaystyle m\times n}
21061:Numerical linear algebra
20900:Newman–Penrose formalism
20688:General purpose software
20551:Numerical linear algebra
19774:Hjørungnes, Are (2011).
19359:
19354:Von Neumann regular ring
18655:{\displaystyle Z=A^{+}B}
18485:{\displaystyle z=A^{+}b}
18310:{\displaystyle I-A^{+}A}
18029:{\displaystyle Z=A^{+}B}
17587:{\displaystyle z=A^{+}b}
15479:{\displaystyle Q^{*}Q=I}
14238:{\displaystyle A^{+}=A.}
9332:converges to the matrix
8664:in the range. Then form
8492:In other words: To find
6628:{\displaystyle Q=A^{+}A}
6592:{\displaystyle P=AA^{+}}
5617:{\displaystyle BB^{+}=I}
5557:{\displaystyle A^{+}A=I}
3000:Existence and uniqueness
2859:is the pseudoinverse of
2679:{\displaystyle AA^{+}=I}
2505:is surjective, and thus
2455:{\displaystyle A^{+}A=I}
20860:Abstract index notation
20496:"Moore–Penrose Inverse"
16368:above), then modifying
16359:{\displaystyle \Sigma }
16269:{\displaystyle \Sigma }
15352:which may be solved by
15183:upper triangular matrix
15011:Consider the case when
14202:{\displaystyle A^{2}=A}
14169:{\displaystyle A=A^{*}}
13791:) or orthonormal rows (
13470:{\displaystyle m\leq n}
13147:{\displaystyle n\leq m}
12115:exists and thus equals
9171:Tikhonov regularization
8089:{\displaystyle \oplus }
7918:{\displaystyle A^{+}=A}
6805:{\displaystyle PA=AQ=A}
6771:). The following hold:
6764:{\displaystyle Q^{2}=Q}
6731:{\displaystyle P^{2}=P}
6698:{\displaystyle Q=Q^{*}}
6665:{\displaystyle P=P^{*}}
5687:{\displaystyle B=A^{+}}
5652:{\displaystyle B=A^{*}}
2465:If, on the other hand,
2281:is injective, and thus
2159:, that is, its rank is
1028:is defined as a matrix
825:{\displaystyle \ker(A)}
319:will denote one of the
21019:John Beresford Leathes
20959:Penrose–Lucas argument
20950:Penrose–Terrell effect
20745:The Emperor's New Mind
19302:
19250:
19198:
19136:
19116:
19089:
18977:
18942:
18911:
18877:
18819:
18795:
18712:
18656:
18620:
18585:
18538:
18486:
18450:
18417:
18381:
18335:
18311:
18271:
18249:
18208:
18184:
18109:
18065:
18030:
17994:
17918:
17867:
17819:
17792:
17742:
17718:
17696:
17621:
17588:
17552:
17472:
17429:
17409:
17389:
17367:
17343:
17309:
17128:
17104:
17103:{\displaystyle A^{*}A}
17078:with full row rank or
17070:
17046:
17045:{\displaystyle AA^{*}}
17012:
16982:
16900:
16835:
16804:
16773:
16749:
16711:
16650:
16607:
16542:
16511:
16487:
16384:
16360:
16336:
16270:
16244:
16189:
16167:
16117:
16015:
15993:
15925:
15896:
15895:{\displaystyle A^{*}A}
15864:
15811:
15780:
15758:
15680:
15679:{\displaystyle A^{*}A}
15646:
15622:
15502:
15480:
15444:
15424:
15389:
15388:{\displaystyle A^{*}A}
15344:
15173:
15151:
15110:Cholesky decomposition
15102:
15027:
15000:
14972:
14971:{\displaystyle A^{*}A}
14938:
14937:{\displaystyle AA^{*}}
14912:computing the product
14906:
14854:
14713:
14689:
14644:
14601:
14566:
14542:
14493:
14438:
14303:
14271:
14239:
14203:
14170:
14135:
14103:
14079:
14057:
14015:
13984:
13960:
13928:
13900:
13871:
13828:
13785:
13740:
13710:
13665:
13649:is a right inverse of
13641:
13610:
13532:
13531:{\displaystyle AA^{*}}
13495:
13471:
13441:
13417:
13387:
13342:
13318:
13287:
13209:
13208:{\displaystyle A^{*}A}
13172:
13148:
13118:
13094:
13064:
12981:
12955:
12928:
12893:
12837:
12811:
12791:
12718:
12683:
12627:
12600:
12400:
12294:
12270:
12246:
12208:
12136:
12100:
12010:
11934:
11838:
11773:
11727:
11648:
11584:
11499:
11429:
11321:
11225:
11198:
11197:{\displaystyle AA^{*}}
11168:
11167:{\displaystyle A^{*}A}
11138:
11065:
10990:
10965:
10858:
10789:
10710:
10607:
10527:
10462:
10409:
10340:
10262:
10206:
10164:
10137:
10110:
10090:
9749:
9722:
9702:
9675:
9631:
9602:
9557:
9526:
9482:
9466:have the same rank as
9458:
9427:
9396:
9348:
9324:
9273:
9219:
9163:
8977:
8953:
8895:
8894:{\displaystyle A^{+}b}
8861:
8837:
8804:
8771:
8747:
8714:
8656:
8623:
8599:
8575:
8542:
8518:
8517:{\displaystyle A^{+}b}
8484:
8436:
8406:
8377:
8316:
8246:
8174:
8146:
8118:
8117:{\displaystyle \perp }
8090:
8066:
8005:
7979:
7929:Geometric construction
7919:
7884:
7860:
7798:
7753:
7705:
7681:
7657:
7635:
7590:
7559:
7496:
7451:
7405:
7300:
7191:
7162:
7110:
7088:
7036:
7012:
6981:
6955:
6920:
6888:
6865:
6806:
6765:
6732:
6699:
6666:
6629:
6593:
6551:
6075:
6012:
5854:
5753:
5688:
5653:
5618:
5580:
5558:
5520:
5496:
5435:
5411:
5350:
5323:
5259:
5107:
5008:
4820:
4666:
4602:
4527:
4456:
4455:{\displaystyle AA^{*}}
4422:
4421:{\displaystyle A^{*}A}
4388:
4316:
4236:
4140:
4120:
4093:
3977:
3957:
3930:
3826:
3758:
3685:
3657:
3589:
3558:
3534:
3352:
3301:
3252:
3225:
3194:
3148:
3095:
3094:{\textstyle AA^{+}A=A}
3051:
3020:
2985:
2965:
2919:
2873:
2853:
2826:
2773:
2753:
2727:
2680:
2640:
2562:
2531:
2530:{\displaystyle AA^{*}}
2499:
2479:
2456:
2416:
2338:
2307:
2306:{\displaystyle A^{*}A}
2273:
2249:
2221:
2190:
2149:
2129:
2094:exists for any matrix
2088:
2058:
1999:
1972:
1971:{\displaystyle A^{+}A}
1942:
1897:
1877:
1876:{\displaystyle AA^{+}}
1847:
1846:{\displaystyle AA^{+}}
1817:
1816:{\displaystyle A^{+}A}
1787:
1767:
1747:
1746:{\displaystyle AA^{+}}
1717:
1697:
1670:
1669:{\displaystyle A^{+}A}
1640:
1581:
1522:
1521:{\displaystyle AA^{+}}
1492:
1491:{\displaystyle A^{+}A}
1459:
1390:
1389:{\displaystyle A^{+}A}
1357:
1284:
1283:{\displaystyle AA^{+}}
1251:
1188:
1158:
1109:
1108:{\displaystyle AA^{+}}
1072:
1018:
966:
911:
881:
854:
826:
790:
766:
730:
694:
648:
608:
576:
541:
510:
462:
423:
397:
367:
341:
311:
270:
246:
215:
180:
80:
53:
20895:Moore–Penrose inverse
20870:Geometry of spacetime
20683:Specialized libraries
20596:Matrix multiplication
20591:Matrix decompositions
20342:Greville, Thomas N.E.
19303:
19251:
19199:
19137:
19117:
19115:{\displaystyle A^{*}}
19090:
18978:
18976:{\displaystyle H_{2}}
18943:
18941:{\displaystyle H_{2}}
18912:
18910:{\displaystyle H_{1}}
18878:
18820:
18796:
18713:
18657:
18621:
18586:
18539:
18487:
18451:
18423:among all solutions.
18418:
18382:
18380:{\displaystyle Ax=b,}
18336:
18312:
18272:
18250:
18209:
18192:for arbitrary vector
18185:
18110:
18084:If the linear system
18066:
18031:
17995:
17919:
17868:
17820:
17793:
17743:
17719:
17697:
17622:
17589:
17553:
17473:
17430:
17410:
17390:
17368:
17351:in general, a vector
17344:
17342:{\displaystyle Ax=b,}
17310:
17199:The MASS package for
17129:
17105:
17071:
17047:
17013:
16983:
16901:
16836:
16834:{\displaystyle A_{0}}
16805:
16803:{\displaystyle A_{i}}
16774:
16750:
16712:
16651:
16608:
16543:
16541:{\displaystyle A_{0}}
16512:
16488:
16415:) uses the recursion
16385:
16361:
16337:
16271:
16245:
16190:
16168:
16118:
16016:
15994:
15926:
15897:
15865:
15812:
15810:{\displaystyle A^{*}}
15781:
15759:
15681:
15647:
15623:
15503:
15481:
15445:
15425:
15390:
15345:
15174:
15152:
15103:
15028:
15008:may be used instead.
15001:
14973:
14939:
14907:
14855:
14714:
14690:
14645:
14602:
14567:
14543:
14494:
14439:
14304:
14272:
14240:
14204:
14171:
14136:
14104:
14080:
14058:
14016:
13985:
13961:
13929:
13901:
13872:
13829:
13786:
13741:
13711:
13666:
13642:
13640:{\displaystyle A^{+}}
13611:
13533:
13496:
13472:
13442:
13418:
13388:
13343:
13319:
13317:{\displaystyle A^{+}}
13288:
13210:
13173:
13149:
13119:
13095:
13065:
12982:
12956:
12954:{\displaystyle A^{+}}
12929:
12894:
12838:
12812:
12797:. More generally, if
12792:
12719:
12684:
12628:
12601:
12401:
12295:
12271:
12247:
12209:
12137:
12135:{\displaystyle A^{+}}
12111:For this matrix, the
12101:
12016:the pseudoinverse is
12011:
11935:
11839:
11774:
11728:
11649:
11585:
11500:
11430:
11322:
11226:
11224:{\displaystyle A^{+}}
11199:
11169:
11139:
11066:
10991:
10966:
10859:
10790:
10711:
10608:
10533:the pseudoinverse is
10528:
10463:
10410:
10346:the pseudoinverse is
10341:
10263:
10207:
10165:
10163:{\displaystyle x_{0}}
10138:
10136:{\displaystyle A^{+}}
10111:
10091:
9750:
9748:{\displaystyle x_{0}}
9723:
9703:
9701:{\displaystyle x_{0}}
9676:
9632:
9630:{\displaystyle x_{0}}
9603:
9558:
9556:{\displaystyle A^{+}}
9527:
9483:
9459:
9457:{\displaystyle A_{n}}
9428:
9426:{\displaystyle A^{+}}
9404:need not converge to
9397:
9349:
9325:
9274:
9220:
9164:
8978:
8954:
8896:
8862:
8838:
8805:
8772:
8748:
8715:
8657:
8624:
8600:
8576:
8543:
8519:
8485:
8437:
8407:
8405:{\displaystyle A^{+}}
8378:
8317:
8247:
8175:
8147:
8145:{\displaystyle \ker }
8128:orthogonal complement
8119:
8091:
8067:
8006:
7980:
7920:
7885:
7861:
7799:
7754:
7706:
7682:
7658:
7636:
7591:
7560:
7497:
7452:
7406:
7301:
7192:
7190:{\displaystyle A^{*}}
7163:
7111:
7089:
7037:
7013:
7011:{\displaystyle A^{*}}
6982:
6956:
6954:{\displaystyle A^{*}}
6930:orthogonal complement
6921:
6889:
6866:
6807:
6766:
6733:
6700:
6667:
6630:
6594:
6552:
6076:
6013:
5855:
5754:
5689:
5654:
5619:
5581:
5559:
5521:
5497:
5436:
5412:
5351:
5324:
5260:
5108:
5009:
4821:
4667:
4603:
4528:
4457:
4423:
4389:
4317:
4237:
4141:
4121:
4119:{\displaystyle A^{*}}
4094:
3978:
3958:
3956:{\displaystyle A^{+}}
3931:
3827:
3759:
3686:
3658:
3590:
3588:{\displaystyle A^{+}}
3559:
3535:
3353:
3302:
3253:
3226:
3224:{\displaystyle A^{+}}
3195:
3149:
3096:
3052:
3050:{\displaystyle A^{+}}
3021:
2986:
2966:
2920:
2874:
2854:
2852:{\displaystyle D^{+}}
2827:
2774:
2754:
2728:
2681:
2641:
2563:
2561:{\displaystyle A^{+}}
2532:
2500:
2480:
2457:
2417:
2339:
2337:{\displaystyle A^{+}}
2308:
2274:
2250:
2222:
2220:{\displaystyle A^{+}}
2191:
2150:
2130:
2089:
2087:{\displaystyle A^{+}}
2059:
2000:
1998:{\displaystyle A^{T}}
1973:
1943:
1898:
1878:
1848:
1818:
1788:
1768:
1748:
1718:
1698:
1696:{\displaystyle A^{T}}
1671:
1646:. More specifically,
1641:
1582:
1523:
1493:
1460:
1391:
1358:
1285:
1252:
1189:
1187:{\displaystyle A^{+}}
1159:
1110:
1073:
1024:, a pseudoinverse of
1019:
967:
912:
882:
855:
827:
791:
767:
731:
695:
649:
609:
577:
575:{\displaystyle A^{*}}
542:
511:
463:
424:
398:
368:
342:
312:
271:
247:
216:
214:{\displaystyle A^{+}}
181:
81:
54:
52:{\displaystyle A^{+}}
29:Moore–Penrose inverse
21025:Illumination problem
20885:Penrose inequalities
20242:Mathematical Gazette
20196:Mathematical Gazette
20133:"LinearAlgebra.pinv"
20023:Gramß, Tino (1992).
19886:-Circulant Matrix".
19393:Rao & Mitra 1971
19260:
19208:
19154:
19126:
19099:
19003:
18960:
18925:
18894:
18841:
18809:
18739:
18666:
18630:
18619:{\displaystyle AX=B}
18601:
18554:
18496:
18460:
18449:{\displaystyle Ax=b}
18431:
18394:
18359:
18345:indeterminate system
18325:
18285:
18261:
18220:
18198:
18123:
18108:{\displaystyle Ax=b}
18090:
18040:
18004:
17930:
17884:
17836:
17806:
17756:
17732:
17708:
17635:
17598:
17562:
17484:
17444:
17419:
17399:
17379:
17357:
17321:
17278:
17254:Linear least-squares
17118:
17084:
17060:
17026:
17002:
16996:For the cases where
16910:
16847:
16818:
16787:
16763:
16723:
16660:
16617:
16554:
16525:
16501:
16419:
16401:Optimized approaches
16374:
16350:
16326:
16260:
16201:
16179:
16138:
16027:
16005:
15935:
15924:{\displaystyle p(t)}
15906:
15876:
15833:
15794:
15770:
15695:
15660:
15636:
15514:
15492:
15454:
15434:
15423:{\displaystyle A=QR}
15405:
15369:
15354:forward substitution
15189:
15163:
15115:
15039:
15017:
14990:
14952:
14918:
14872:
14725:
14703:
14658:
14613:
14600:{\displaystyle A=BC}
14582:
14556:
14511:
14462:
14319:
14289:
14261:
14213:
14180:
14147:
14125:
14093:
14069:
14025:
14014:{\displaystyle p(t)}
13996:
13974:
13950:
13918:
13890:
13838:
13795:
13752:
13730:
13677:
13655:
13624:
13544:
13512:
13504:linearly independent
13485:
13455:
13431:
13407:
13354:
13332:
13301:
13221:
13189:
13181:linearly independent
13162:
13132:
13108:
13084:
12991:
12965:
12938:
12903:
12847:
12821:
12801:
12728:
12693:
12637:
12617:
12419:
12306:
12284:
12260:
12236:
12146:
12119:
12020:
11946:
11847:
11786:
11737:
11657:
11596:
11508:
11444:
11331:
11238:
11208:
11178:
11148:
11075:
11002:
10980:
10868:
10799:
10720:
10622:
10537:
10475:
10419:
10350:
10288:
10216:
10174:
10147:
10120:
10100:
10096:where the functions
9759:
9732:
9712:
9685:
9643:
9639:. The derivative of
9614:
9577:
9540:
9496:
9472:
9441:
9410:
9366:
9338:
9299:
9233:
9179:
9004:
8967:
8909:
8875:
8851:
8818:
8803:{\displaystyle p(b)}
8785:
8761:
8728:
8670:
8655:{\displaystyle p(b)}
8637:
8613:
8589:
8556:
8532:
8498:
8448:
8420:
8389:
8326:
8256:
8186:
8164:
8136:
8108:
8080:
8019:
7993:
7939:
7896:
7874:
7809:
7767:
7722:
7695:
7671:
7647:
7641:, and checking that
7600:
7589:{\displaystyle C=BA}
7571:
7508:
7465:
7420:
7311:
7210:
7174:
7124:
7100:
7050:
7026:
6995:
6971:
6938:
6910:
6898:orthogonal projector
6878:
6816:
6778:
6742:
6709:
6676:
6643:
6603:
6567:
6089:
6026:
5867:
5768:
5704:
5665:
5630:
5592:
5570:
5532:
5510:
5447:
5425:
5362:
5340:
5274:
5118:
5019:
4831:
4677:
4617:
4541:
4478:
4436:
4402:
4325:
4253:
4150:
4130:
4103:
3987:
3967:
3940:
3845:
3768:
3700:
3669:
3601:
3572:
3548:
3365:
3314:
3268:
3242:
3208:
3184:
3105:
3066:
3034:
3010:
2975:
2929:
2883:
2863:
2836:
2783:
2763:
2737:
2733:for some isometries
2698:
2654:
2574:
2545:
2511:
2489:
2469:
2430:
2350:
2321:
2287:
2263:
2239:
2204:
2165:
2139:
2098:
2071:
2009:
1982:
1952:
1907:
1887:
1857:
1827:
1797:
1777:
1757:
1727:
1707:
1680:
1650:
1591:
1532:
1502:
1472:
1402:
1370:
1300:
1264:
1204:
1171:
1125:
1089:
1034:
987:
928:
895:
871:
844:
804:
780:
756:
708:
663:
618:
588:
559:
524:
479:
437:
411:
381:
355:
329:
299:
260:
236:
198:
170:
70:
36:
23:, and in particular
20761:The Road to Reality
20753:Shadows of the Mind
20570:Numerical stability
20162:1956PCPS...52...17P
19994:1974SJNA...11...61S
19948:1966SJNA....3..410B
19745:1973SJNA...10..413G
19631:1966SIAMR...8..518G
19537:Matrix computations
19531:Charles F. Van Loan
19494:1955PCPS...51..406P
19317:*-regular semigroup
19148:identity involution
18731:, one can define a
18355:For linear systems
17395:is in the image of
17172:The Python package
17159:numerical expertise
16697:
13035:
11316:
4099:while substituting
2752:{\displaystyle U,V}
1398:is also Hermitian:
551:conjugate transpose
123:generalized inverse
116:in 1903. The terms
88:, often called the
20493:Weisstein, Eric W.
20474:Weisstein, Eric W.
20398:. Addison-Wesley.
20093:SIAM J. Appl. Math
20066:SIAM J. Appl. Math
19801:Statistical Papers
19714:Matematički Vesnik
19349:Pseudo-determinant
19298:
19246:
19194:
19182:
19132:
19112:
19085:
18973:
18938:
18907:
18873:
18815:
18791:
18747:
18718:for all solutions.
18708:
18652:
18616:
18581:
18544:for all solutions.
18534:
18482:
18446:
18413:
18377:
18331:
18307:
18267:
18245:
18204:
18180:
18105:
18061:
18026:
17990:
17914:
17863:
17818:{\displaystyle z.}
17815:
17788:
17738:
17714:
17692:
17617:
17584:
17548:
17468:
17425:
17405:
17385:
17363:
17339:
17305:
17167:embedded computing
17163:parallel computing
17149:Software libraries
17124:
17100:
17066:
17042:
17008:
16978:
16896:
16831:
16800:
16769:
16745:
16707:
16683:
16646:
16603:
16538:
16507:
16483:
16380:
16356:
16332:
16266:
16240:
16185:
16163:
16113:
16011:
15989:
15921:
15892:
15860:
15807:
15776:
15754:
15676:
15642:
15618:
15498:
15476:
15440:
15420:
15385:
15340:
15169:
15147:
15098:
15023:
14996:
14968:
14934:
14902:
14850:
14709:
14685:
14640:
14597:
14562:
14538:
14489:
14452:Rank decomposition
14434:
14432:
14299:
14267:
14248:Circulant matrices
14235:
14199:
14166:
14131:
14099:
14075:
14053:
14011:
13980:
13956:
13944:A (square) matrix
13924:
13896:
13867:
13824:
13781:
13736:
13706:
13661:
13637:
13606:
13528:
13491:
13467:
13437:
13413:
13383:
13338:
13314:
13283:
13205:
13168:
13144:
13114:
13090:
13060:
13018:
12977:
12951:
12924:
12889:
12833:
12807:
12787:
12714:
12679:
12623:
12596:
12591:
12396:
12391:
12384:
12343:
12290:
12266:
12242:
12204:
12195:
12132:
12096:
12090:
12006:
11997:
11930:
11921:
11834:
11825:
11769:
11723:
11717:
11644:
11635:
11580:
11571:
11495:
11486:
11425:
11317:
11300:
11221:
11194:
11164:
11134:
11061:
10986:
10961:
10942:
10854:
10848:
10785:
10773:
10706:
10700:
10603:
10597:
10523:
10514:
10458:
10405:
10396:
10336:
10327:
10258:
10202:
10160:
10133:
10106:
10086:
9745:
9718:
9698:
9671:
9627:
9598:
9553:
9522:
9478:
9454:
9423:
9392:
9344:
9320:
9293:: if the sequence
9269:
9215:
9159:
9106:
9035:
8973:
8949:
8891:
8857:
8833:
8800:
8767:
8743:
8710:
8652:
8631:, finding a point
8619:
8595:
8571:
8538:
8514:
8480:
8432:
8402:
8373:
8322:. The restriction
8312:
8242:
8170:
8142:
8114:
8086:
8062:
8001:
7975:
7915:
7880:
7856:
7794:
7749:
7701:
7677:
7653:
7631:
7586:
7555:
7492:
7447:
7401:
7296:
7187:
7158:
7106:
7084:
7032:
7008:
6977:
6951:
6928:(which equals the
6916:
6884:
6861:
6802:
6761:
6728:
6705:) and idempotent (
6695:
6662:
6625:
6589:
6547:
6535:
6492:
6452:
6443:
6422:
6398:
6394:
6373:
6341:
6332:
6311:
6282:
6273:
6252:
6219:
6165:
6129:
6071:
6008:
6006:
5850:
5749:
5684:
5649:
5614:
5576:
5554:
5516:
5492:
5431:
5407:
5346:
5319:
5255:
5253:
5103:
5004:
5002:
4816:
4814:
4662:
4598:
4523:
4452:
4418:
4384:
4312:
4232:
4136:
4116:
4089:
3973:
3953:
3926:
3822:
3754:
3681:
3653:
3585:
3554:
3530:
3348:
3297:
3248:
3221:
3190:
3144:
3091:
3047:
3016:
2981:
2961:
2915:
2869:
2849:
2822:
2769:
2749:
2723:
2676:
2636:
2570:can be computed as
2558:
2527:
2495:
2475:
2452:
2412:
2346:can be computed as
2334:
2303:
2269:
2245:
2217:
2186:
2145:
2125:
2084:
2067:The pseudoinverse
2054:
1995:
1968:
1938:
1893:
1873:
1843:
1813:
1783:
1763:
1743:
1713:
1693:
1666:
1636:
1577:
1518:
1488:
1455:
1386:
1353:
1280:
1247:
1184:
1154:
1105:
1068:
1014:
962:
907:
877:
850:
822:
786:
762:
726:
690:
644:
604:
572:
537:
506:
458:
419:
393:
363:
337:
307:
266:
242:
211:
176:
114:integral operators
110:Erik Ivar Fredholm
108:in 1955. Earlier,
76:
49:
21038:
21037:
20974:Andromeda paradox
20945:Penrose transform
20875:Cosmic censorship
20696:
20695:
20432:978-0-471-70821-6
20386:978-0-486-66693-8
20363:978-0-387-00293-4
19598:978-0-387-95452-3
19552:978-0-8018-5414-9
19135:{\displaystyle A}
18818:{\displaystyle A}
18746:
18334:{\displaystyle A}
18270:{\displaystyle A}
18207:{\displaystyle w}
17741:{\displaystyle A}
17717:{\displaystyle w}
17428:{\displaystyle A}
17408:{\displaystyle A}
17388:{\displaystyle b}
17366:{\displaystyle x}
17194:scipy.linalg.pinv
17127:{\displaystyle A}
17069:{\displaystyle A}
17011:{\displaystyle A}
16772:{\displaystyle A}
16510:{\displaystyle A}
16383:{\displaystyle A}
16335:{\displaystyle A}
16305:, where ε is the
16188:{\displaystyle A}
16014:{\displaystyle A}
15827:For an arbitrary
15779:{\displaystyle A}
15645:{\displaystyle R}
15501:{\displaystyle R}
15443:{\displaystyle Q}
15358:back substitution
15172:{\displaystyle R}
15026:{\displaystyle A}
14999:{\displaystyle A}
14712:{\displaystyle r}
14576:(rank) decomposed
14565:{\displaystyle A}
14281:Fourier transform
14270:{\displaystyle C}
14134:{\displaystyle A}
14102:{\displaystyle A}
14078:{\displaystyle A}
13983:{\displaystyle A}
13959:{\displaystyle A}
13927:{\displaystyle A}
13899:{\displaystyle A}
13739:{\displaystyle A}
13664:{\displaystyle A}
13494:{\displaystyle m}
13440:{\displaystyle m}
13416:{\displaystyle A}
13341:{\displaystyle A}
13171:{\displaystyle n}
13117:{\displaystyle n}
13093:{\displaystyle A}
12810:{\displaystyle A}
12754:
12705:
12655:
12626:{\displaystyle D}
12609:Diagonal matrices
12584:
12570:
12552:
12525:
12505:
12490:
12479:
12462:
12432:
12383:
12342:
12293:{\displaystyle x}
12269:{\displaystyle x}
12245:{\displaystyle x}
12086:
12074:
11917:
11905:
11891:
11879:
11701:
11689:
11555:
11540:
11413:
11412:
11357:
11356:
11294:
11293:
11252:
10989:{\displaystyle A}
10926:
10914:
10696:
10684:
10670:
10658:
10581:
10569:
10109:{\displaystyle A}
10054:
9991:
9985:
9948:
9891:
9885:
9867:
9783:
9721:{\displaystyle A}
9534:will converge to
9481:{\displaystyle A}
9347:{\displaystyle A}
9091:
9020:
8976:{\displaystyle A}
8860:{\displaystyle A}
8770:{\displaystyle A}
8622:{\displaystyle A}
8598:{\displaystyle b}
8541:{\displaystyle b}
7883:{\displaystyle A}
7704:{\displaystyle A}
7680:{\displaystyle C}
7656:{\displaystyle D}
7289:
7286:
7219:
7109:{\displaystyle A}
7035:{\displaystyle A}
6980:{\displaystyle Q}
6932:of the kernel of
6919:{\displaystyle A}
6887:{\displaystyle P}
6442:
6421:
6393:
6372:
6331:
6310:
6272:
6251:
5579:{\displaystyle B}
5519:{\displaystyle A}
5434:{\displaystyle B}
5349:{\displaystyle A}
4573:
4139:{\displaystyle A}
3976:{\displaystyle A}
3557:{\displaystyle A}
3468:
3438:
3251:{\displaystyle A}
3193:{\displaystyle A}
3154:, it is called a
3019:{\displaystyle A}
2984:{\displaystyle A}
2872:{\displaystyle D}
2772:{\displaystyle D}
2498:{\displaystyle A}
2478:{\displaystyle A}
2272:{\displaystyle A}
2248:{\displaystyle A}
2148:{\displaystyle A}
2135:. If furthermore
1896:{\displaystyle A}
1786:{\displaystyle A}
1766:{\displaystyle A}
1716:{\displaystyle A}
880:{\displaystyle n}
853:{\displaystyle A}
789:{\displaystyle A}
765:{\displaystyle A}
269:{\displaystyle A}
245:{\displaystyle A}
179:{\displaystyle A}
79:{\displaystyle A}
21068:
21007:Jonathan Penrose
20964:FELIX experiment
20930:Penrose triangle
20835:
20823:
20820:Nancy Cartwright
20805:
20788:Coauthored books
20723:
20716:
20709:
20700:
20699:
20606:Matrix splitting
20544:
20537:
20530:
20521:
20520:
20506:
20505:
20487:
20486:
20436:
20420:
20409:
20390:
20378:
20367:
20325:
20324:
20314:
20290:
20284:
20283:
20275:
20266:
20265:
20237:
20228:
20227:
20191:
20182:
20181:
20143:
20137:
20136:
20129:
20123:
20122:
20115:
20109:
20108:
20088:
20082:
20081:
20061:
20055:
20054:
20052:
20043:
20037:
20036:
20020:
20014:
20013:
19977:
19971:
19967:
19931:
19925:
19920:
19914:
19913:
19880:Stallings, W. T.
19876:
19870:
19869:
19867:
19837:
19828:
19822:
19813:
19812:
19796:
19790:
19789:
19771:
19765:
19764:
19728:
19722:
19721:
19711:
19702:
19696:
19695:
19685:
19657:
19651:
19650:
19610:
19604:
19602:
19580:
19569:
19563:
19557:
19556:
19540:
19523:
19508:
19507:
19505:
19468:
19459:
19458:
19447:Bjerhammar, Arne
19443:
19437:
19436:
19434:
19405:
19399:
19369:
19313:abstract algebra
19307:
19305:
19304:
19299:
19291:
19287:
19283:
19282:
19255:
19253:
19252:
19247:
19239:
19235:
19234:
19233:
19203:
19201:
19200:
19195:
19193:
19192:
19187:
19186:
19141:
19139:
19138:
19133:
19121:
19119:
19118:
19113:
19111:
19110:
19094:
19092:
19091:
19086:
19084:
19080:
19079:
19078:
19052:
19048:
19044:
19043:
18984:
18982:
18980:
18979:
18974:
18972:
18971:
18949:
18947:
18945:
18944:
18939:
18937:
18936:
18918:
18916:
18914:
18913:
18908:
18906:
18905:
18884:
18882:
18880:
18879:
18874:
18872:
18871:
18859:
18858:
18826:
18824:
18822:
18821:
18816:
18800:
18798:
18797:
18792:
18787:
18783:
18782:
18748:
18744:
18735:for any matrix:
18733:condition number
18723:Condition number
18717:
18715:
18714:
18709:
18707:
18706:
18705:
18686:
18685:
18684:
18661:
18659:
18658:
18653:
18648:
18647:
18625:
18623:
18622:
18617:
18592:
18590:
18588:
18587:
18582:
18580:
18579:
18568:
18543:
18541:
18540:
18535:
18533:
18532:
18514:
18513:
18491:
18489:
18488:
18483:
18478:
18477:
18455:
18453:
18452:
18447:
18422:
18420:
18419:
18414:
18412:
18411:
18386:
18384:
18383:
18378:
18342:
18340:
18338:
18337:
18332:
18318:
18316:
18314:
18313:
18308:
18303:
18302:
18278:
18276:
18274:
18273:
18268:
18254:
18252:
18251:
18246:
18235:
18234:
18215:
18213:
18211:
18210:
18205:
18189:
18187:
18186:
18181:
18176:
18172:
18168:
18167:
18141:
18140:
18114:
18112:
18111:
18106:
18070:
18068:
18067:
18062:
18060:
18059:
18058:
18035:
18033:
18032:
18027:
18022:
18021:
17999:
17997:
17996:
17991:
17989:
17988:
17987:
17959:
17958:
17957:
17925:
17923:
17921:
17920:
17915:
17913:
17912:
17901:
17874:
17872:
17870:
17869:
17864:
17862:
17861:
17850:
17826:
17824:
17822:
17821:
17816:
17799:
17797:
17795:
17794:
17789:
17787:
17783:
17779:
17778:
17749:
17747:
17745:
17744:
17739:
17725:
17723:
17721:
17720:
17715:
17701:
17699:
17698:
17693:
17688:
17684:
17680:
17679:
17653:
17652:
17626:
17624:
17623:
17618:
17616:
17615:
17593:
17591:
17590:
17585:
17580:
17579:
17557:
17555:
17554:
17549:
17547:
17546:
17541:
17537:
17515:
17514:
17509:
17505:
17479:
17477:
17475:
17474:
17469:
17467:
17466:
17461:
17434:
17432:
17431:
17426:
17414:
17412:
17411:
17406:
17394:
17392:
17391:
17386:
17374:
17372:
17370:
17369:
17364:
17348:
17346:
17345:
17340:
17316:
17314:
17312:
17311:
17306:
17304:
17303:
17292:
17244:
17233:
17232:pseudo_inverse()
17229:
17218:
17214:
17210:
17206:
17195:
17192:adds a function
17187:
17183:
17179:
17135:
17133:
17131:
17130:
17125:
17111:
17109:
17107:
17106:
17101:
17096:
17095:
17077:
17075:
17073:
17072:
17067:
17053:
17051:
17049:
17048:
17043:
17041:
17040:
17019:
17017:
17015:
17014:
17009:
16987:
16985:
16984:
16979:
16977:
16976:
16967:
16966:
16958:
16954:
16941:
16940:
16922:
16921:
16905:
16903:
16902:
16897:
16895:
16894:
16889:
16885:
16881:
16880:
16859:
16858:
16842:
16840:
16838:
16837:
16832:
16830:
16829:
16811:
16809:
16807:
16806:
16801:
16799:
16798:
16780:
16778:
16776:
16775:
16770:
16756:
16754:
16752:
16751:
16746:
16735:
16734:
16716:
16714:
16713:
16708:
16696:
16691:
16682:
16655:
16653:
16652:
16647:
16645:
16644:
16629:
16628:
16612:
16610:
16609:
16604:
16602:
16601:
16596:
16592:
16588:
16587:
16566:
16565:
16549:
16547:
16545:
16544:
16539:
16537:
16536:
16518:
16516:
16514:
16513:
16508:
16492:
16490:
16489:
16484:
16479:
16478:
16466:
16465:
16453:
16452:
16437:
16436:
16391:
16389:
16387:
16386:
16381:
16367:
16365:
16363:
16362:
16357:
16343:
16341:
16339:
16338:
16333:
16304:
16289:
16277:
16275:
16273:
16272:
16267:
16249:
16247:
16246:
16241:
16239:
16238:
16229:
16228:
16213:
16212:
16196:
16194:
16192:
16191:
16186:
16172:
16170:
16169:
16164:
16162:
16161:
16122:
16120:
16119:
16114:
16106:
16105:
16087:
16086:
16074:
16073:
16058:
16057:
16039:
16038:
16022:
16020:
16018:
16017:
16012:
15998:
15996:
15995:
15990:
15982:
15981:
15963:
15962:
15950:
15949:
15930:
15928:
15927:
15922:
15901:
15899:
15898:
15893:
15888:
15887:
15871:
15869:
15867:
15866:
15861:
15859:
15858:
15847:
15818:
15816:
15814:
15813:
15808:
15806:
15805:
15787:
15785:
15783:
15782:
15777:
15763:
15761:
15760:
15755:
15753:
15752:
15744:
15740:
15739:
15738:
15720:
15719:
15707:
15706:
15687:
15685:
15683:
15682:
15677:
15672:
15671:
15653:
15651:
15649:
15648:
15643:
15627:
15625:
15624:
15619:
15611:
15610:
15590:
15589:
15580:
15579:
15553:
15552:
15526:
15525:
15509:
15507:
15505:
15504:
15499:
15485:
15483:
15482:
15477:
15466:
15465:
15449:
15447:
15446:
15441:
15429:
15427:
15426:
15421:
15399:QR decomposition
15396:
15394:
15392:
15391:
15386:
15381:
15380:
15349:
15347:
15346:
15341:
15339:
15338:
15326:
15325:
15313:
15312:
15298:
15297:
15285:
15284:
15275:
15271:
15267:
15266:
15247:
15246:
15237:
15236:
15228:
15224:
15220:
15219:
15201:
15200:
15180:
15178:
15176:
15175:
15170:
15156:
15154:
15153:
15148:
15143:
15142:
15127:
15126:
15107:
15105:
15104:
15099:
15097:
15096:
15087:
15086:
15078:
15074:
15070:
15069:
15051:
15050:
15034:
15032:
15030:
15029:
15024:
15007:
15005:
15003:
15002:
14997:
14982:QR decomposition
14979:
14977:
14975:
14974:
14969:
14964:
14963:
14945:
14943:
14941:
14940:
14935:
14933:
14932:
14911:
14909:
14908:
14903:
14898:
14890:
14879:
14859:
14857:
14856:
14851:
14849:
14848:
14839:
14838:
14830:
14826:
14822:
14821:
14806:
14805:
14797:
14793:
14792:
14791:
14773:
14772:
14760:
14759:
14750:
14749:
14737:
14736:
14720:
14718:
14716:
14715:
14710:
14696:
14694:
14692:
14691:
14686:
14684:
14683:
14672:
14651:
14649:
14647:
14646:
14641:
14639:
14638:
14627:
14606:
14604:
14603:
14598:
14573:
14571:
14569:
14568:
14563:
14549:
14547:
14545:
14544:
14539:
14537:
14536:
14525:
14500:
14498:
14496:
14495:
14490:
14443:
14441:
14440:
14435:
14433:
14426:
14425:
14420:
14419:
14409:
14408:
14396:
14395:
14382:
14381:
14365:
14364:
14359:
14358:
14342:
14341:
14310:
14308:
14306:
14305:
14300:
14298:
14297:
14278:
14276:
14274:
14273:
14268:
14254:circulant matrix
14244:
14242:
14241:
14236:
14225:
14224:
14208:
14206:
14205:
14200:
14192:
14191:
14175:
14173:
14172:
14167:
14165:
14164:
14142:
14140:
14138:
14137:
14132:
14110:
14108:
14106:
14105:
14100:
14086:
14084:
14082:
14081:
14076:
14062:
14060:
14059:
14054:
14037:
14036:
14020:
14018:
14017:
14012:
13991:
13989:
13987:
13986:
13981:
13967:
13965:
13963:
13962:
13957:
13935:
13933:
13931:
13930:
13925:
13907:
13905:
13903:
13902:
13897:
13876:
13874:
13873:
13868:
13863:
13862:
13850:
13849:
13833:
13831:
13830:
13825:
13823:
13822:
13810:
13809:
13790:
13788:
13787:
13782:
13780:
13779:
13764:
13763:
13747:
13745:
13743:
13742:
13737:
13715:
13713:
13712:
13707:
13705:
13704:
13692:
13691:
13672:
13670:
13668:
13667:
13662:
13648:
13646:
13644:
13643:
13638:
13636:
13635:
13618:It follows that
13615:
13613:
13612:
13607:
13602:
13601:
13593:
13589:
13588:
13587:
13569:
13568:
13556:
13555:
13539:
13537:
13535:
13534:
13529:
13527:
13526:
13502:
13500:
13498:
13497:
13492:
13478:
13476:
13474:
13473:
13468:
13448:
13446:
13444:
13443:
13438:
13424:
13422:
13420:
13419:
13414:
13392:
13390:
13389:
13384:
13382:
13381:
13366:
13365:
13349:
13347:
13345:
13344:
13339:
13325:
13323:
13321:
13320:
13315:
13313:
13312:
13295:It follows that
13292:
13290:
13289:
13284:
13279:
13278:
13269:
13268:
13260:
13256:
13252:
13251:
13233:
13232:
13216:
13214:
13212:
13211:
13206:
13201:
13200:
13179:
13177:
13175:
13174:
13169:
13155:
13153:
13151:
13150:
13145:
13125:
13123:
13121:
13120:
13115:
13101:
13099:
13097:
13096:
13091:
13069:
13067:
13066:
13061:
13059:
13058:
13046:
13034:
13029:
13006:
13005:
12986:
12984:
12983:
12978:
12960:
12958:
12957:
12952:
12950:
12949:
12933:
12931:
12930:
12925:
12923:
12915:
12914:
12898:
12896:
12895:
12890:
12888:
12887:
12878:
12877:
12862:
12861:
12842:
12840:
12839:
12834:
12816:
12814:
12813:
12808:
12796:
12794:
12793:
12788:
12786:
12785:
12774:
12765:
12764:
12756:
12755:
12747:
12740:
12739:
12723:
12721:
12720:
12715:
12707:
12706:
12698:
12688:
12686:
12685:
12680:
12678:
12677:
12666:
12657:
12656:
12648:
12632:
12630:
12629:
12624:
12605:
12603:
12602:
12597:
12595:
12594:
12585:
12582:
12572:
12571:
12563:
12560:
12559:
12554:
12553:
12545:
12538:
12533:
12532:
12527:
12526:
12518:
12507:
12506:
12498:
12492:
12491:
12483:
12480:
12477:
12470:
12469:
12464:
12463:
12455:
12440:
12439:
12434:
12433:
12425:
12405:
12403:
12402:
12397:
12395:
12394:
12385:
12381:
12373:
12372:
12344:
12340:
12318:
12317:
12301:
12299:
12297:
12296:
12291:
12277:
12275:
12273:
12272:
12267:
12253:
12251:
12249:
12248:
12243:
12213:
12211:
12210:
12205:
12200:
12199:
12158:
12157:
12141:
12139:
12138:
12133:
12131:
12130:
12105:
12103:
12102:
12097:
12095:
12094:
12087:
12079:
12075:
12067:
12032:
12031:
12015:
12013:
12012:
12007:
12002:
12001:
11939:
11937:
11936:
11931:
11926:
11925:
11918:
11910:
11906:
11898:
11892:
11884:
11880:
11872:
11859:
11858:
11843:
11841:
11840:
11835:
11830:
11829:
11778:
11776:
11775:
11770:
11768:
11767:
11755:
11754:
11732:
11730:
11729:
11724:
11722:
11721:
11702:
11694:
11690:
11682:
11669:
11668:
11653:
11651:
11650:
11645:
11640:
11639:
11589:
11587:
11586:
11581:
11576:
11575:
11556:
11548:
11541:
11533:
11520:
11519:
11504:
11502:
11501:
11496:
11491:
11490:
11434:
11432:
11431:
11426:
11424:
11423:
11418:
11414:
11408:
11407:
11406:
11405:
11400:
11391:
11390:
11385:
11378:
11371:
11370:
11365:
11358:
11352:
11348:
11343:
11342:
11326:
11324:
11323:
11318:
11315:
11310:
11305:
11299:
11295:
11289:
11288:
11287:
11286:
11281:
11272:
11271:
11266:
11259:
11253:
11248:
11230:
11228:
11227:
11222:
11220:
11219:
11203:
11201:
11200:
11195:
11193:
11192:
11173:
11171:
11170:
11165:
11160:
11159:
11143:
11141:
11140:
11135:
11133:
11132:
11124:
11120:
11119:
11118:
11100:
11099:
11087:
11086:
11070:
11068:
11067:
11062:
11060:
11059:
11050:
11049:
11041:
11037:
11033:
11032:
11014:
11013:
10997:
10995:
10993:
10992:
10987:
10970:
10968:
10967:
10962:
10960:
10959:
10947:
10946:
10927:
10919:
10915:
10907:
10894:
10893:
10880:
10879:
10863:
10861:
10860:
10855:
10853:
10852:
10811:
10810:
10794:
10792:
10791:
10786:
10778:
10777:
10736:
10735:
10715:
10713:
10712:
10707:
10705:
10704:
10697:
10689:
10685:
10677:
10671:
10663:
10659:
10651:
10638:
10637:
10612:
10610:
10609:
10604:
10602:
10601:
10582:
10574:
10570:
10562:
10549:
10548:
10532:
10530:
10529:
10524:
10519:
10518:
10467:
10465:
10464:
10459:
10457:
10456:
10444:
10443:
10431:
10430:
10414:
10412:
10411:
10406:
10401:
10400:
10362:
10361:
10345:
10343:
10342:
10337:
10332:
10331:
10272:is established.
10267:
10265:
10264:
10259:
10254:
10253:
10241:
10240:
10228:
10227:
10211:
10209:
10208:
10203:
10198:
10197:
10169:
10167:
10166:
10161:
10159:
10158:
10142:
10140:
10139:
10134:
10132:
10131:
10115:
10113:
10112:
10107:
10095:
10093:
10092:
10087:
10082:
10081:
10072:
10071:
10059:
10055:
10053:
10049:
10043:
10042:
10041:
10032:
10026:
10020:
10016:
10012:
10011:
9989:
9983:
9982:
9978:
9977:
9976:
9953:
9949:
9947:
9943:
9937:
9936:
9935:
9926:
9920:
9914:
9913:
9901:
9900:
9889:
9883:
9882:
9881:
9872:
9868:
9866:
9862:
9856:
9852:
9846:
9840:
9839:
9824:
9823:
9814:
9813:
9805:
9804:
9788:
9784:
9782:
9778:
9772:
9767:
9754:
9752:
9751:
9746:
9744:
9743:
9727:
9725:
9724:
9719:
9707:
9705:
9704:
9699:
9697:
9696:
9680:
9678:
9677:
9672:
9661:
9660:
9638:
9636:
9634:
9633:
9628:
9626:
9625:
9607:
9605:
9604:
9599:
9564:
9562:
9560:
9559:
9554:
9552:
9551:
9533:
9531:
9529:
9528:
9523:
9521:
9520:
9511:
9510:
9489:
9487:
9485:
9484:
9479:
9465:
9463:
9461:
9460:
9455:
9453:
9452:
9434:
9432:
9430:
9429:
9424:
9422:
9421:
9403:
9401:
9399:
9398:
9393:
9391:
9390:
9381:
9380:
9355:
9353:
9351:
9350:
9345:
9331:
9329:
9327:
9326:
9321:
9319:
9315:
9314:
9280:
9278:
9276:
9275:
9270:
9268:
9267:
9259:
9255:
9251:
9250:
9226:
9224:
9222:
9221:
9216:
9214:
9213:
9205:
9201:
9200:
9199:
9168:
9166:
9165:
9160:
9158:
9157:
9149:
9145:
9135:
9134:
9116:
9115:
9105:
9087:
9086:
9077:
9076:
9068:
9064:
9051:
9050:
9034:
9016:
9015:
8984:
8982:
8980:
8979:
8974:
8960:
8958:
8956:
8955:
8950:
8924:
8923:
8902:
8900:
8898:
8897:
8892:
8887:
8886:
8868:
8866:
8864:
8863:
8858:
8844:
8842:
8840:
8839:
8834:
8832:
8831:
8826:
8811:
8809:
8807:
8806:
8801:
8778:
8776:
8774:
8773:
8768:
8754:
8752:
8750:
8749:
8744:
8742:
8741:
8736:
8721:
8719:
8717:
8716:
8711:
8685:
8684:
8663:
8661:
8659:
8658:
8653:
8630:
8628:
8626:
8625:
8620:
8606:
8604:
8602:
8601:
8596:
8583:, first project
8582:
8580:
8578:
8577:
8572:
8570:
8569:
8564:
8549:
8547:
8545:
8544:
8539:
8525:
8523:
8521:
8520:
8515:
8510:
8509:
8489:
8487:
8486:
8481:
8476:
8475:
8470:
8466:
8443:
8441:
8439:
8438:
8433:
8413:
8411:
8409:
8408:
8403:
8401:
8400:
8382:
8380:
8379:
8374:
8360:
8359:
8354:
8350:
8321:
8319:
8318:
8313:
8311:
8310:
8305:
8301:
8270:
8269:
8264:
8251:
8249:
8248:
8243:
8229:
8228:
8223:
8219:
8200:
8199:
8194:
8181:
8179:
8177:
8176:
8171:
8153:
8151:
8149:
8148:
8143:
8125:
8123:
8121:
8120:
8115:
8097:
8095:
8093:
8092:
8087:
8073:
8071:
8069:
8068:
8063:
8061:
8060:
8055:
8046:
8045:
8040:
8031:
8030:
8012:
8010:
8008:
8007:
8002:
8000:
7986:
7984:
7982:
7981:
7976:
7974:
7973:
7968:
7959:
7958:
7953:
7924:
7922:
7921:
7916:
7908:
7907:
7891:
7889:
7887:
7886:
7881:
7865:
7863:
7862:
7857:
7855:
7854:
7830:
7829:
7805:
7803:
7801:
7800:
7795:
7793:
7792:
7781:
7760:
7758:
7756:
7755:
7750:
7748:
7747:
7736:
7712:
7710:
7708:
7707:
7702:
7688:
7686:
7684:
7683:
7678:
7664:
7662:
7660:
7659:
7654:
7640:
7638:
7637:
7632:
7630:
7629:
7595:
7593:
7592:
7587:
7564:
7562:
7561:
7556:
7554:
7553:
7532:
7531:
7503:
7501:
7499:
7498:
7493:
7491:
7490:
7479:
7458:
7456:
7454:
7453:
7448:
7446:
7445:
7434:
7410:
7408:
7407:
7402:
7394:
7393:
7384:
7380:
7376:
7375:
7352:
7348:
7347:
7346:
7323:
7322:
7305:
7303:
7302:
7297:
7287:
7284:
7280:
7276:
7275:
7274:
7248:
7244:
7240:
7239:
7217:
7198:
7196:
7194:
7193:
7188:
7186:
7185:
7167:
7165:
7164:
7159:
7157:
7156:
7117:
7115:
7113:
7112:
7107:
7093:
7091:
7090:
7085:
7080:
7079:
7043:
7041:
7039:
7038:
7033:
7019:
7017:
7015:
7014:
7009:
7007:
7006:
6988:
6986:
6984:
6983:
6978:
6962:
6960:
6958:
6957:
6952:
6950:
6949:
6927:
6925:
6923:
6922:
6917:
6895:
6893:
6891:
6890:
6885:
6870:
6868:
6867:
6862:
6860:
6859:
6847:
6846:
6828:
6827:
6811:
6809:
6808:
6803:
6770:
6768:
6767:
6762:
6754:
6753:
6737:
6735:
6734:
6729:
6721:
6720:
6704:
6702:
6701:
6696:
6694:
6693:
6671:
6669:
6668:
6663:
6661:
6660:
6634:
6632:
6631:
6626:
6621:
6620:
6598:
6596:
6595:
6590:
6588:
6587:
6556:
6554:
6553:
6548:
6546:
6545:
6540:
6539:
6503:
6502:
6497:
6496:
6457:
6456:
6444:
6435:
6423:
6414:
6403:
6402:
6395:
6386:
6374:
6365:
6346:
6345:
6333:
6324:
6312:
6303:
6287:
6286:
6274:
6265:
6253:
6244:
6230:
6229:
6224:
6223:
6184:
6183:
6178:
6177:
6170:
6169:
6134:
6133:
6098:
6097:
6082:
6080:
6078:
6077:
6072:
6070:
6069:
6060:
6059:
6047:
6046:
6017:
6015:
6014:
6009:
6007:
6000:
5999:
5987:
5986:
5970:
5969:
5964:
5960:
5956:
5955:
5933:
5932:
5923:
5922:
5903:
5902:
5897:
5893:
5892:
5891:
5859:
5857:
5856:
5851:
5843:
5842:
5827:
5826:
5805:
5804:
5783:
5782:
5760:
5758:
5756:
5755:
5750:
5748:
5747:
5738:
5737:
5725:
5724:
5693:
5691:
5690:
5685:
5683:
5682:
5658:
5656:
5655:
5650:
5648:
5647:
5623:
5621:
5620:
5615:
5607:
5606:
5587:
5585:
5583:
5582:
5577:
5563:
5561:
5560:
5555:
5544:
5543:
5527:
5525:
5523:
5522:
5517:
5501:
5499:
5498:
5493:
5491:
5490:
5478:
5477:
5462:
5461:
5442:
5440:
5438:
5437:
5432:
5416:
5414:
5413:
5408:
5406:
5405:
5390:
5389:
5374:
5373:
5357:
5355:
5353:
5352:
5347:
5330:
5328:
5326:
5325:
5320:
5318:
5317:
5308:
5307:
5295:
5294:
5264:
5262:
5261:
5256:
5254:
5247:
5246:
5222:
5221:
5205:
5204:
5195:
5194:
5169:
5168:
5134:
5133:
5112:
5110:
5109:
5104:
5099:
5098:
5089:
5088:
5073:
5072:
5057:
5056:
5047:
5046:
5031:
5030:
5013:
5011:
5010:
5005:
5003:
4996:
4995:
4980:
4979:
4963:
4962:
4957:
4953:
4952:
4951:
4936:
4935:
4913:
4912:
4897:
4896:
4880:
4879:
4874:
4870:
4869:
4868:
4853:
4852:
4825:
4823:
4822:
4817:
4815:
4802:
4801:
4779:
4778:
4769:
4768:
4749:
4748:
4739:
4738:
4719:
4718:
4709:
4708:
4693:
4692:
4671:
4669:
4668:
4663:
4661:
4660:
4651:
4650:
4638:
4637:
4609:
4607:
4605:
4604:
4599:
4597:
4596:
4585:
4571:
4567:
4566:
4555:
4534:
4532:
4530:
4529:
4524:
4522:
4521:
4512:
4511:
4499:
4498:
4463:
4461:
4459:
4458:
4453:
4451:
4450:
4429:
4427:
4425:
4424:
4419:
4414:
4413:
4393:
4391:
4390:
4385:
4380:
4379:
4374:
4370:
4369:
4368:
4350:
4349:
4337:
4336:
4321:
4319:
4318:
4313:
4308:
4307:
4298:
4297:
4292:
4288:
4284:
4283:
4265:
4264:
4241:
4239:
4238:
4233:
4228:
4227:
4218:
4213:
4211:
4210:
4201:
4196:
4194:
4193:
4184:
4179:
4177:
4176:
4167:
4162:
4161:
4145:
4143:
4142:
4137:
4125:
4123:
4122:
4117:
4115:
4114:
4098:
4096:
4095:
4090:
4085:
4084:
4075:
4073:
4072:
4060:
4058:
4057:
4048:
4043:
4041:
4040:
4031:
4029:
4028:
4016:
4014:
4013:
4004:
3999:
3998:
3982:
3980:
3979:
3974:
3962:
3960:
3959:
3954:
3952:
3951:
3935:
3933:
3932:
3927:
3919:
3917:
3916:
3907:
3905:
3904:
3892:
3887:
3885:
3884:
3872:
3870:
3869:
3860:
3855:
3831:
3829:
3828:
3823:
3821:
3817:
3816:
3794:
3790:
3789:
3763:
3761:
3760:
3755:
3753:
3749:
3748:
3726:
3722:
3721:
3692:
3690:
3688:
3687:
3682:
3662:
3660:
3659:
3654:
3652:
3651:
3642:
3641:
3626:
3625:
3620:
3616:
3596:
3594:
3592:
3591:
3586:
3584:
3583:
3565:
3563:
3561:
3560:
3555:
3539:
3537:
3536:
3531:
3526:
3525:
3520:
3516:
3515:
3498:
3497:
3492:
3488:
3487:
3469:
3464:
3463:
3454:
3449:
3448:
3443:
3439:
3431:
3420:
3419:
3414:
3410:
3409:
3392:
3391:
3386:
3382:
3381:
3357:
3355:
3354:
3349:
3341:
3340:
3335:
3331:
3330:
3306:
3304:
3303:
3298:
3296:
3295:
3280:
3279:
3259:
3257:
3255:
3254:
3249:
3232:
3230:
3228:
3227:
3222:
3220:
3219:
3201:
3199:
3197:
3196:
3191:
3166:Basic properties
3153:
3151:
3150:
3145:
3143:
3142:
3130:
3129:
3117:
3116:
3100:
3098:
3097:
3092:
3081:
3080:
3058:
3056:
3054:
3053:
3048:
3046:
3045:
3027:
3025:
3023:
3022:
3017:
2991:, respectively.
2990:
2988:
2987:
2982:
2970:
2968:
2967:
2962:
2960:
2959:
2941:
2940:
2924:
2922:
2921:
2916:
2914:
2913:
2898:
2897:
2878:
2876:
2875:
2870:
2858:
2856:
2855:
2850:
2848:
2847:
2831:
2829:
2828:
2823:
2821:
2820:
2811:
2810:
2795:
2794:
2778:
2776:
2775:
2770:
2758:
2756:
2755:
2750:
2732:
2730:
2729:
2724:
2722:
2721:
2685:
2683:
2682:
2677:
2669:
2668:
2645:
2643:
2642:
2637:
2632:
2631:
2623:
2619:
2618:
2617:
2599:
2598:
2586:
2585:
2569:
2567:
2565:
2564:
2559:
2557:
2556:
2539:is invertible),
2538:
2536:
2534:
2533:
2528:
2526:
2525:
2504:
2502:
2501:
2496:
2484:
2482:
2481:
2476:
2461:
2459:
2458:
2453:
2442:
2441:
2421:
2419:
2418:
2413:
2408:
2407:
2398:
2397:
2389:
2385:
2381:
2380:
2362:
2361:
2345:
2343:
2341:
2340:
2335:
2333:
2332:
2315:is invertible),
2314:
2312:
2310:
2309:
2304:
2299:
2298:
2280:
2278:
2276:
2275:
2270:
2256:
2254:
2252:
2251:
2246:
2228:
2226:
2224:
2223:
2218:
2216:
2215:
2197:
2195:
2193:
2192:
2187:
2154:
2152:
2151:
2146:
2134:
2132:
2131:
2126:
2124:
2123:
2112:
2093:
2091:
2090:
2085:
2083:
2082:
2063:
2061:
2060:
2055:
2053:
2052:
2040:
2039:
2024:
2023:
2004:
2002:
2001:
1996:
1994:
1993:
1977:
1975:
1974:
1969:
1964:
1963:
1947:
1945:
1944:
1939:
1925:
1924:
1902:
1900:
1899:
1894:
1882:
1880:
1879:
1874:
1872:
1871:
1852:
1850:
1849:
1844:
1842:
1841:
1822:
1820:
1819:
1814:
1809:
1808:
1792:
1790:
1789:
1784:
1772:
1770:
1769:
1764:
1752:
1750:
1749:
1744:
1742:
1741:
1722:
1720:
1719:
1714:
1702:
1700:
1699:
1694:
1692:
1691:
1675:
1673:
1672:
1667:
1662:
1661:
1645:
1643:
1642:
1637:
1632:
1631:
1619:
1618:
1606:
1605:
1586:
1584:
1583:
1578:
1576:
1575:
1560:
1559:
1550:
1549:
1527:
1525:
1524:
1519:
1517:
1516:
1497:
1495:
1494:
1489:
1484:
1483:
1464:
1462:
1461:
1456:
1448:
1447:
1434:
1433:
1428:
1424:
1420:
1419:
1397:
1395:
1393:
1392:
1387:
1382:
1381:
1362:
1360:
1359:
1354:
1349:
1348:
1332:
1331:
1326:
1322:
1321:
1320:
1291:
1289:
1287:
1286:
1281:
1279:
1278:
1256:
1254:
1253:
1248:
1243:
1242:
1229:
1228:
1216:
1215:
1195:
1193:
1191:
1190:
1185:
1183:
1182:
1163:
1161:
1160:
1155:
1140:
1139:
1120:
1116:
1114:
1112:
1111:
1106:
1104:
1103:
1079:
1077:
1075:
1074:
1069:
1067:
1066:
1055:
1046:
1045:
1027:
1023:
1021:
1020:
1015:
1013:
1012:
1001:
973:
971:
969:
968:
963:
961:
960:
949:
940:
939:
918:
916:
914:
913:
908:
888:
886:
884:
883:
878:
861:
859:
857:
856:
851:
838:(null space) of
833:
831:
829:
828:
823:
797:
795:
793:
792:
787:
773:
771:
769:
768:
763:
737:
735:
733:
732:
727:
701:
699:
697:
696:
691:
689:
688:
677:
653:
651:
650:
645:
643:
642:
630:
629:
613:
611:
610:
605:
603:
595:
583:
581:
579:
578:
573:
571:
570:
548:
546:
544:
543:
538:
536:
535:
517:
515:
513:
512:
507:
505:
504:
493:
469:
467:
465:
464:
459:
457:
456:
445:
430:
428:
426:
425:
420:
418:
404:
402:
400:
399:
394:
374:
372:
370:
369:
364:
362:
348:
346:
344:
343:
338:
336:
318:
316:
314:
313:
308:
306:
278:on the subspace
277:
275:
273:
272:
267:
253:
251:
249:
248:
243:
222:
220:
218:
217:
212:
210:
209:
187:
185:
183:
182:
177:
87:
85:
83:
82:
77:
60:
58:
56:
55:
50:
48:
47:
21076:
21075:
21071:
21070:
21069:
21067:
21066:
21065:
21041:
21040:
21039:
21034:
21013:Shirley Hodgson
20983:
20969:Trapped surface
20920:Penrose process
20905:Penrose diagram
20865:Black hole bomb
20838:
20832:Brian W. Aldiss
20826:
20808:
20802:Stephen Hawking
20794:
20783:
20732:
20727:
20697:
20692:
20651:
20647:Multiprocessing
20615:
20611:Sparse problems
20574:
20553:
20548:
20477:"Pseudoinverse"
20443:
20433:
20406:
20387:
20364:
20338:Ben-Israel, Adi
20333:
20328:
20291:
20287:
20276:
20269:
20248:(420): 109–14.
20238:
20231:
20208:10.2307/3617890
20202:(425): 181–85.
20192:
20185:
20144:
20140:
20131:
20130:
20126:
20117:
20116:
20112:
20105:10.1137/0124033
20089:
20085:
20078:10.1137/0125057
20062:
20058:
20050:
20044:
20040:
20021:
20017:
20002:10.1137/0711008
19978:
19974:
19956:10.1137/0703035
19932:
19928:
19921:
19917:
19902:10.2307/2038377
19877:
19873:
19838:
19831:
19823:
19816:
19797:
19793:
19786:
19772:
19768:
19753:10.1137/0710036
19729:
19725:
19709:
19703:
19699:
19658:
19654:
19639:10.1137/1008107
19611:
19607:
19599:
19589:Springer-Verlag
19581:
19572:
19564:
19560:
19553:
19524:
19511:
19469:
19462:
19444:
19440:
19406:
19402:
19398:
19395:, p. 50–51
19370:
19366:
19362:
19339:Inverse element
19325:
19278:
19274:
19273:
19269:
19261:
19258:
19257:
19229:
19225:
19221:
19217:
19209:
19206:
19205:
19204:. Observe that
19188:
19181:
19180:
19175:
19165:
19164:
19163:
19155:
19152:
19151:
19127:
19124:
19123:
19106:
19102:
19100:
19097:
19096:
19074:
19070:
19066:
19062:
19039:
19035:
19034:
19030:
19004:
19001:
19000:
18967:
18963:
18961:
18958:
18957:
18955:
18932:
18928:
18926:
18923:
18922:
18920:
18901:
18897:
18895:
18892:
18891:
18889:
18867:
18863:
18854:
18850:
18842:
18839:
18838:
18836:
18833:
18831:Generalizations
18810:
18807:
18806:
18804:
18778:
18774:
18770:
18742:
18740:
18737:
18736:
18725:
18701:
18700:
18696:
18680:
18679:
18675:
18667:
18664:
18663:
18643:
18639:
18631:
18628:
18627:
18602:
18599:
18598:
18569:
18564:
18563:
18555:
18552:
18551:
18549:
18528:
18524:
18509:
18505:
18497:
18494:
18493:
18473:
18469:
18461:
18458:
18457:
18432:
18429:
18428:
18407:
18403:
18395:
18392:
18391:
18360:
18357:
18356:
18353:
18326:
18323:
18322:
18320:
18298:
18294:
18286:
18283:
18282:
18280:
18262:
18259:
18258:
18256:
18230:
18226:
18221:
18218:
18217:
18199:
18196:
18195:
18193:
18163:
18159:
18152:
18148:
18136:
18132:
18124:
18121:
18120:
18091:
18088:
18087:
18082:
18054:
18053:
18049:
18041:
18038:
18037:
18017:
18013:
18005:
18002:
18001:
17983:
17982:
17978:
17953:
17952:
17948:
17931:
17928:
17927:
17902:
17897:
17896:
17885:
17882:
17881:
17879:
17851:
17846:
17845:
17837:
17834:
17833:
17831:
17807:
17804:
17803:
17801:
17774:
17770:
17763:
17759:
17757:
17754:
17753:
17751:
17733:
17730:
17729:
17727:
17709:
17706:
17705:
17703:
17702:for any vector
17675:
17671:
17664:
17660:
17648:
17644:
17636:
17633:
17632:
17611:
17607:
17599:
17596:
17595:
17575:
17571:
17563:
17560:
17559:
17542:
17524:
17520:
17519:
17510:
17492:
17488:
17487:
17485:
17482:
17481:
17462:
17457:
17456:
17445:
17442:
17441:
17439:
17420:
17417:
17416:
17400:
17397:
17396:
17380:
17377:
17376:
17358:
17355:
17354:
17352:
17322:
17319:
17318:
17293:
17288:
17287:
17279:
17276:
17275:
17273:
17262:
17256:
17251:
17242:
17231:
17227:
17216:
17212:
17208:
17204:
17193:
17185:
17181:
17177:
17151:
17119:
17116:
17115:
17113:
17091:
17087:
17085:
17082:
17081:
17079:
17061:
17058:
17057:
17055:
17036:
17032:
17027:
17024:
17023:
17021:
17003:
17000:
16999:
16997:
16994:
16972:
16968:
16959:
16936:
16932:
16931:
16927:
16926:
16917:
16913:
16911:
16908:
16907:
16890:
16876:
16872:
16871:
16867:
16866:
16854:
16850:
16848:
16845:
16844:
16825:
16821:
16819:
16816:
16815:
16813:
16794:
16790:
16788:
16785:
16784:
16782:
16764:
16761:
16760:
16758:
16730:
16726:
16724:
16721:
16720:
16718:
16692:
16687:
16678:
16661:
16658:
16657:
16640:
16636:
16624:
16620:
16618:
16615:
16614:
16597:
16583:
16579:
16578:
16574:
16573:
16561:
16557:
16555:
16552:
16551:
16532:
16528:
16526:
16523:
16522:
16520:
16502:
16499:
16498:
16496:
16474:
16470:
16461:
16457:
16448:
16444:
16426:
16422:
16420:
16417:
16416:
16409:
16398:
16375:
16372:
16371:
16369:
16351:
16348:
16347:
16345:
16327:
16324:
16323:
16321:
16307:machine epsilon
16291:
16287:
16261:
16258:
16257:
16255:
16234:
16230:
16224:
16220:
16208:
16204:
16202:
16199:
16198:
16180:
16177:
16176:
16174:
16157:
16153:
16139:
16136:
16135:
16128:
16101:
16097:
16082:
16078:
16069:
16065:
16053:
16049:
16034:
16030:
16028:
16025:
16024:
16006:
16003:
16002:
16000:
15977:
15973:
15958:
15954:
15945:
15941:
15936:
15933:
15932:
15907:
15904:
15903:
15883:
15879:
15877:
15874:
15873:
15872:, one has that
15848:
15843:
15842:
15834:
15831:
15830:
15828:
15825:
15801:
15797:
15795:
15792:
15791:
15789:
15771:
15768:
15767:
15765:
15745:
15734:
15730:
15726:
15722:
15721:
15715:
15711:
15702:
15698:
15696:
15693:
15692:
15667:
15663:
15661:
15658:
15657:
15655:
15637:
15634:
15633:
15631:
15606:
15602:
15585:
15581:
15575:
15571:
15548:
15544:
15521:
15517:
15515:
15512:
15511:
15493:
15490:
15489:
15487:
15461:
15457:
15455:
15452:
15451:
15435:
15432:
15431:
15406:
15403:
15402:
15376:
15372:
15370:
15367:
15366:
15364:
15334:
15330:
15321:
15317:
15308:
15304:
15293:
15289:
15280:
15276:
15262:
15258:
15257:
15253:
15242:
15238:
15229:
15215:
15211:
15210:
15206:
15205:
15196:
15192:
15190:
15187:
15186:
15164:
15161:
15160:
15158:
15138:
15134:
15122:
15118:
15116:
15113:
15112:
15092:
15088:
15079:
15065:
15061:
15060:
15056:
15055:
15046:
15042:
15040:
15037:
15036:
15018:
15015:
15014:
15012:
14991:
14988:
14987:
14985:
14959:
14955:
14953:
14950:
14949:
14947:
14928:
14924:
14919:
14916:
14915:
14913:
14894:
14886:
14875:
14873:
14870:
14869:
14866:
14844:
14840:
14831:
14817:
14813:
14812:
14808:
14807:
14798:
14787:
14783:
14779:
14775:
14774:
14768:
14764:
14755:
14751:
14745:
14741:
14732:
14728:
14726:
14723:
14722:
14704:
14701:
14700:
14698:
14673:
14668:
14667:
14659:
14656:
14655:
14653:
14628:
14623:
14622:
14614:
14611:
14610:
14608:
14583:
14580:
14579:
14557:
14554:
14553:
14551:
14526:
14521:
14520:
14512:
14509:
14508:
14506:
14463:
14460:
14459:
14457:
14454:
14449:
14431:
14430:
14421:
14415:
14414:
14413:
14404:
14400:
14391:
14390:
14383:
14377:
14373:
14370:
14369:
14360:
14354:
14353:
14352:
14337:
14336:
14329:
14322:
14320:
14317:
14316:
14293:
14292:
14290:
14287:
14286:
14284:
14262:
14259:
14258:
14256:
14250:
14220:
14216:
14214:
14211:
14210:
14187:
14183:
14181:
14178:
14177:
14160:
14156:
14148:
14145:
14144:
14126:
14123:
14122:
14120:
14117:
14094:
14091:
14090:
14088:
14070:
14067:
14066:
14064:
14032:
14028:
14026:
14023:
14022:
13997:
13994:
13993:
13992:. A polynomial
13975:
13972:
13971:
13969:
13951:
13948:
13947:
13945:
13942:
13919:
13916:
13915:
13913:
13891:
13888:
13887:
13885:
13882:
13880:Normal matrices
13858:
13854:
13845:
13841:
13839:
13836:
13835:
13818:
13814:
13805:
13801:
13796:
13793:
13792:
13775:
13771:
13759:
13755:
13753:
13750:
13749:
13731:
13728:
13727:
13725:
13722:
13700:
13696:
13687:
13683:
13678:
13675:
13674:
13656:
13653:
13652:
13650:
13631:
13627:
13625:
13622:
13621:
13619:
13594:
13583:
13579:
13575:
13571:
13570:
13564:
13560:
13551:
13547:
13545:
13542:
13541:
13522:
13518:
13513:
13510:
13509:
13507:
13486:
13483:
13482:
13480:
13456:
13453:
13452:
13450:
13432:
13429:
13428:
13426:
13408:
13405:
13404:
13402:
13401:If the rank of
13399:
13377:
13373:
13361:
13357:
13355:
13352:
13351:
13333:
13330:
13329:
13327:
13308:
13304:
13302:
13299:
13298:
13296:
13274:
13270:
13261:
13247:
13243:
13242:
13238:
13237:
13228:
13224:
13222:
13219:
13218:
13196:
13192:
13190:
13187:
13186:
13184:
13163:
13160:
13159:
13157:
13133:
13130:
13129:
13127:
13109:
13106:
13105:
13103:
13085:
13082:
13081:
13079:
13078:If the rank of
13076:
13051:
13047:
13042:
13030:
13022:
12998:
12994:
12992:
12989:
12988:
12966:
12963:
12962:
12945:
12941:
12939:
12936:
12935:
12919:
12910:
12906:
12904:
12901:
12900:
12883:
12879:
12870:
12866:
12854:
12850:
12848:
12845:
12844:
12822:
12819:
12818:
12802:
12799:
12798:
12775:
12770:
12769:
12757:
12746:
12745:
12744:
12735:
12731:
12729:
12726:
12725:
12697:
12696:
12694:
12691:
12690:
12667:
12662:
12661:
12647:
12646:
12638:
12635:
12634:
12618:
12615:
12614:
12611:
12590:
12589:
12581:
12579:
12562:
12561:
12555:
12544:
12543:
12542:
12534:
12528:
12517:
12516:
12515:
12512:
12511:
12497:
12496:
12482:
12481:
12476:
12474:
12465:
12454:
12453:
12452:
12445:
12444:
12435:
12424:
12423:
12422:
12420:
12417:
12416:
12411:
12390:
12389:
12379:
12377:
12365:
12361:
12358:
12357:
12338:
12336:
12323:
12322:
12313:
12309:
12307:
12304:
12303:
12285:
12282:
12281:
12279:
12261:
12258:
12257:
12255:
12237:
12234:
12233:
12231:
12228:
12223:
12217:
12194:
12193:
12188:
12182:
12181:
12176:
12166:
12165:
12153:
12149:
12147:
12144:
12143:
12126:
12122:
12120:
12117:
12116:
12089:
12088:
12078:
12076:
12066:
12064:
12058:
12057:
12052:
12047:
12037:
12036:
12027:
12023:
12021:
12018:
12017:
11996:
11995:
11990:
11984:
11983:
11978:
11972:
11971:
11966:
11956:
11955:
11947:
11944:
11943:
11920:
11919:
11909:
11907:
11897:
11894:
11893:
11883:
11881:
11871:
11864:
11863:
11854:
11850:
11848:
11845:
11844:
11824:
11823:
11818:
11812:
11811:
11806:
11796:
11795:
11787:
11784:
11783:
11763:
11759:
11750:
11746:
11738:
11735:
11734:
11716:
11715:
11710:
11704:
11703:
11693:
11691:
11681:
11674:
11673:
11664:
11660:
11658:
11655:
11654:
11634:
11633:
11628:
11622:
11621:
11616:
11606:
11605:
11597:
11594:
11593:
11570:
11569:
11564:
11558:
11557:
11547:
11542:
11532:
11525:
11524:
11515:
11511:
11509:
11506:
11505:
11485:
11484:
11479:
11470:
11469:
11464:
11454:
11453:
11445:
11442:
11441:
11419:
11401:
11396:
11395:
11386:
11381:
11380:
11379:
11377:
11373:
11372:
11366:
11361:
11360:
11347:
11338:
11334:
11332:
11329:
11328:
11311:
11306:
11301:
11282:
11277:
11276:
11267:
11262:
11261:
11260:
11258:
11254:
11247:
11239:
11236:
11235:
11215:
11211:
11209:
11206:
11205:
11188:
11184:
11179:
11176:
11175:
11155:
11151:
11149:
11146:
11145:
11125:
11114:
11110:
11106:
11102:
11101:
11095:
11091:
11082:
11078:
11076:
11073:
11072:
11055:
11051:
11042:
11028:
11024:
11023:
11019:
11018:
11009:
11005:
11003:
11000:
10999:
10981:
10978:
10977:
10975:
10955:
10951:
10941:
10940:
10935:
10929:
10928:
10918:
10916:
10906:
10899:
10898:
10889:
10885:
10875:
10871:
10869:
10866:
10865:
10847:
10846:
10841:
10835:
10834:
10829:
10819:
10818:
10806:
10802:
10800:
10797:
10796:
10772:
10771:
10766:
10760:
10759:
10754:
10744:
10743:
10731:
10727:
10721:
10718:
10717:
10699:
10698:
10688:
10686:
10676:
10673:
10672:
10662:
10660:
10650:
10643:
10642:
10633:
10629:
10623:
10620:
10619:
10596:
10595:
10590:
10584:
10583:
10573:
10571:
10561:
10554:
10553:
10544:
10540:
10538:
10535:
10534:
10513:
10512:
10507:
10501:
10500:
10495:
10485:
10484:
10476:
10473:
10472:
10452:
10448:
10439:
10435:
10426:
10422:
10420:
10417:
10416:
10395:
10394:
10389:
10383:
10382:
10377:
10367:
10366:
10357:
10353:
10351:
10348:
10347:
10326:
10325:
10320:
10314:
10313:
10308:
10298:
10297:
10289:
10286:
10285:
10278:
10249:
10245:
10236:
10232:
10223:
10219:
10217:
10214:
10213:
10193:
10189:
10175:
10172:
10171:
10154:
10150:
10148:
10145:
10144:
10127:
10123:
10121:
10118:
10117:
10101:
10098:
10097:
10077:
10073:
10064:
10060:
10045:
10044:
10037:
10033:
10028:
10027:
10025:
10021:
10007:
10003:
9996:
9992:
9972:
9968:
9958:
9954:
9939:
9938:
9931:
9927:
9922:
9921:
9919:
9915:
9906:
9902:
9896:
9892:
9877:
9873:
9858:
9857:
9848:
9847:
9845:
9841:
9835:
9831:
9819:
9815:
9800:
9796:
9789:
9774:
9773:
9768:
9766:
9763:
9762:
9760:
9757:
9756:
9739:
9735:
9733:
9730:
9729:
9713:
9710:
9709:
9692:
9688:
9686:
9683:
9682:
9656:
9652:
9644:
9641:
9640:
9621:
9617:
9615:
9612:
9611:
9609:
9578:
9575:
9574:
9571:
9547:
9543:
9541:
9538:
9537:
9535:
9516:
9512:
9506:
9502:
9497:
9494:
9493:
9491:
9473:
9470:
9469:
9467:
9448:
9444:
9442:
9439:
9438:
9436:
9417:
9413:
9411:
9408:
9407:
9405:
9386:
9382:
9376:
9372:
9367:
9364:
9363:
9361:
9339:
9336:
9335:
9333:
9310:
9306:
9302:
9300:
9297:
9296:
9294:
9287:
9260:
9246:
9242:
9241:
9237:
9236:
9234:
9231:
9230:
9228:
9206:
9195:
9191:
9187:
9183:
9182:
9180:
9177:
9176:
9174:
9150:
9130:
9126:
9122:
9118:
9117:
9111:
9107:
9095:
9082:
9078:
9069:
9046:
9042:
9041:
9037:
9036:
9024:
9011:
9007:
9005:
9002:
9001:
8998:
8996:Limit relations
8968:
8965:
8964:
8962:
8916:
8912:
8910:
8907:
8906:
8904:
8882:
8878:
8876:
8873:
8872:
8870:
8852:
8849:
8848:
8846:
8827:
8822:
8821:
8819:
8816:
8815:
8813:
8786:
8783:
8782:
8780:
8762:
8759:
8758:
8756:
8737:
8732:
8731:
8729:
8726:
8725:
8723:
8677:
8673:
8671:
8668:
8667:
8665:
8638:
8635:
8634:
8632:
8614:
8611:
8610:
8608:
8590:
8587:
8586:
8584:
8565:
8560:
8559:
8557:
8554:
8553:
8551:
8533:
8530:
8529:
8527:
8505:
8501:
8499:
8496:
8495:
8493:
8471:
8456:
8452:
8451:
8449:
8446:
8445:
8421:
8418:
8417:
8415:
8396:
8392:
8390:
8387:
8386:
8384:
8355:
8340:
8336:
8335:
8327:
8324:
8323:
8306:
8291:
8287:
8286:
8265:
8260:
8259:
8257:
8254:
8253:
8224:
8209:
8205:
8204:
8195:
8190:
8189:
8187:
8184:
8183:
8165:
8162:
8161:
8159:
8137:
8134:
8133:
8131:
8109:
8106:
8105:
8103:
8081:
8078:
8077:
8075:
8056:
8051:
8050:
8041:
8036:
8035:
8026:
8022:
8020:
8017:
8016:
8014:
7996:
7994:
7991:
7990:
7988:
7987:over the field
7969:
7964:
7963:
7954:
7949:
7948:
7940:
7937:
7936:
7934:
7931:
7903:
7899:
7897:
7894:
7893:
7875:
7872:
7871:
7869:
7850:
7846:
7825:
7821:
7810:
7807:
7806:
7782:
7777:
7776:
7768:
7765:
7764:
7762:
7737:
7732:
7731:
7723:
7720:
7719:
7717:
7696:
7693:
7692:
7690:
7672:
7669:
7668:
7666:
7648:
7645:
7644:
7642:
7625:
7621:
7601:
7598:
7597:
7572:
7569:
7568:
7549:
7545:
7527:
7523:
7509:
7506:
7505:
7480:
7475:
7474:
7466:
7463:
7462:
7460:
7435:
7430:
7429:
7421:
7418:
7417:
7415:
7389:
7385:
7371:
7367:
7360:
7356:
7342:
7338:
7328:
7324:
7318:
7314:
7312:
7309:
7308:
7270:
7266:
7256:
7252:
7235:
7231:
7224:
7220:
7211:
7208:
7207:
7181:
7177:
7175:
7172:
7171:
7169:
7152:
7148:
7125:
7122:
7121:
7101:
7098:
7097:
7095:
7075:
7071:
7051:
7048:
7047:
7027:
7024:
7023:
7021:
7002:
6998:
6996:
6993:
6992:
6990:
6972:
6969:
6968:
6966:
6945:
6941:
6939:
6936:
6935:
6933:
6911:
6908:
6907:
6905:
6879:
6876:
6875:
6873:
6855:
6851:
6842:
6838:
6823:
6819:
6817:
6814:
6813:
6779:
6776:
6775:
6749:
6745:
6743:
6740:
6739:
6716:
6712:
6710:
6707:
6706:
6689:
6685:
6677:
6674:
6673:
6656:
6652:
6644:
6641:
6640:
6616:
6612:
6604:
6601:
6600:
6583:
6579:
6568:
6565:
6564:
6562:
6541:
6534:
6533:
6528:
6522:
6521:
6516:
6506:
6505:
6504:
6498:
6491:
6490:
6485:
6479:
6478:
6473:
6463:
6462:
6461:
6451:
6450:
6445:
6433:
6430:
6429:
6424:
6412:
6405:
6404:
6397:
6396:
6384:
6382:
6376:
6375:
6363:
6361:
6351:
6350:
6340:
6339:
6334:
6322:
6319:
6318:
6313:
6301:
6294:
6293:
6281:
6280:
6275:
6263:
6260:
6259:
6254:
6242:
6235:
6234:
6225:
6218:
6217:
6212:
6206:
6205:
6200:
6190:
6189:
6188:
6179:
6173:
6172:
6171:
6164:
6163:
6158:
6152:
6151:
6146:
6136:
6135:
6128:
6127:
6122:
6116:
6115:
6110:
6100:
6099:
6093:
6092:
6090:
6087:
6086:
6065:
6061:
6055:
6051:
6042:
6038:
6027:
6024:
6023:
6021:
6005:
6004:
5992:
5988:
5982:
5978:
5971:
5965:
5951:
5947:
5946:
5942:
5941:
5938:
5937:
5928:
5924:
5915:
5911:
5904:
5898:
5887:
5883:
5879:
5875:
5874:
5870:
5868:
5865:
5864:
5838:
5834:
5822:
5818:
5800:
5796:
5778:
5774:
5769:
5766:
5765:
5743:
5739:
5733:
5729:
5720:
5716:
5705:
5702:
5701:
5699:
5678:
5674:
5666:
5663:
5662:
5643:
5639:
5631:
5628:
5627:
5602:
5598:
5593:
5590:
5589:
5571:
5568:
5567:
5565:
5539:
5535:
5533:
5530:
5529:
5511:
5508:
5507:
5505:
5486:
5482:
5473:
5469:
5457:
5453:
5448:
5445:
5444:
5426:
5423:
5422:
5420:
5401:
5397:
5385:
5381:
5369:
5365:
5363:
5360:
5359:
5341:
5338:
5337:
5335:
5313:
5309:
5303:
5299:
5290:
5286:
5275:
5272:
5271:
5269:
5252:
5251:
5242:
5238:
5217:
5213:
5206:
5200:
5196:
5190:
5186:
5180:
5179:
5164:
5160:
5141:
5129:
5125:
5121:
5119:
5116:
5115:
5094:
5090:
5084:
5080:
5068:
5064:
5052:
5048:
5042:
5038:
5026:
5022:
5020:
5017:
5016:
5001:
5000:
4991:
4987:
4975:
4971:
4964:
4958:
4947:
4943:
4931:
4927:
4926:
4922:
4921:
4918:
4917:
4908:
4904:
4892:
4888:
4881:
4875:
4864:
4860:
4848:
4844:
4843:
4839:
4838:
4834:
4832:
4829:
4828:
4813:
4812:
4797:
4793:
4786:
4774:
4770:
4764:
4760:
4754:
4753:
4744:
4740:
4734:
4730:
4720:
4714:
4710:
4704:
4700:
4688:
4684:
4680:
4678:
4675:
4674:
4656:
4652:
4646:
4642:
4633:
4629:
4618:
4615:
4614:
4586:
4581:
4580:
4556:
4551:
4550:
4542:
4539:
4538:
4536:
4517:
4513:
4507:
4503:
4494:
4490:
4479:
4476:
4475:
4473:
4470:
4464:are Hermitian.
4446:
4442:
4437:
4434:
4433:
4431:
4409:
4405:
4403:
4400:
4399:
4397:
4375:
4364:
4360:
4356:
4352:
4351:
4345:
4341:
4332:
4328:
4326:
4323:
4322:
4303:
4299:
4293:
4279:
4275:
4274:
4270:
4269:
4260:
4256:
4254:
4251:
4250:
4247:
4223:
4219:
4217:
4212:
4206:
4202:
4200:
4195:
4189:
4185:
4183:
4178:
4172:
4168:
4166:
4157:
4153:
4151:
4148:
4147:
4131:
4128:
4127:
4110:
4106:
4104:
4101:
4100:
4080:
4076:
4074:
4065:
4061:
4059:
4053:
4049:
4047:
4042:
4036:
4032:
4030:
4021:
4017:
4015:
4009:
4005:
4003:
3994:
3990:
3988:
3985:
3984:
3968:
3965:
3964:
3947:
3943:
3941:
3938:
3937:
3918:
3912:
3908:
3906:
3897:
3893:
3891:
3886:
3877:
3873:
3871:
3865:
3861:
3859:
3854:
3846:
3843:
3842:
3839:
3812:
3808:
3804:
3785:
3781:
3777:
3769:
3766:
3765:
3744:
3740:
3736:
3717:
3713:
3709:
3701:
3698:
3697:
3670:
3667:
3666:
3664:
3647:
3643:
3634:
3630:
3621:
3609:
3605:
3604:
3602:
3599:
3598:
3579:
3575:
3573:
3570:
3569:
3567:
3549:
3546:
3545:
3543:
3521:
3511:
3507:
3503:
3502:
3493:
3483:
3479:
3475:
3474:
3459:
3455:
3453:
3444:
3430:
3426:
3425:
3415:
3405:
3401:
3397:
3396:
3387:
3377:
3373:
3369:
3368:
3366:
3363:
3362:
3336:
3326:
3322:
3318:
3317:
3315:
3312:
3311:
3288:
3284:
3275:
3271:
3269:
3266:
3265:
3243:
3240:
3239:
3237:
3215:
3211:
3209:
3206:
3205:
3203:
3185:
3182:
3181:
3179:
3168:
3138:
3134:
3125:
3121:
3112:
3108:
3106:
3103:
3102:
3076:
3072:
3067:
3064:
3063:
3041:
3037:
3035:
3032:
3031:
3029:
3011:
3008:
3007:
3005:
3002:
2997:
2976:
2973:
2972:
2955:
2951:
2936:
2932:
2930:
2927:
2926:
2909:
2905:
2893:
2889:
2884:
2881:
2880:
2864:
2861:
2860:
2843:
2839:
2837:
2834:
2833:
2816:
2812:
2806:
2802:
2790:
2786:
2784:
2781:
2780:
2764:
2761:
2760:
2738:
2735:
2734:
2717:
2713:
2699:
2696:
2695:
2664:
2660:
2655:
2652:
2651:
2624:
2613:
2609:
2605:
2601:
2600:
2594:
2590:
2581:
2577:
2575:
2572:
2571:
2552:
2548:
2546:
2543:
2542:
2540:
2521:
2517:
2512:
2509:
2508:
2506:
2490:
2487:
2486:
2470:
2467:
2466:
2437:
2433:
2431:
2428:
2427:
2403:
2399:
2390:
2376:
2372:
2371:
2367:
2366:
2357:
2353:
2351:
2348:
2347:
2328:
2324:
2322:
2319:
2318:
2316:
2294:
2290:
2288:
2285:
2284:
2282:
2264:
2261:
2260:
2258:
2240:
2237:
2236:
2234:
2211:
2207:
2205:
2202:
2201:
2199:
2166:
2163:
2162:
2160:
2140:
2137:
2136:
2113:
2108:
2107:
2099:
2096:
2095:
2078:
2074:
2072:
2069:
2068:
2048:
2044:
2035:
2031:
2019:
2015:
2010:
2007:
2006:
1989:
1985:
1983:
1980:
1979:
1959:
1955:
1953:
1950:
1949:
1920:
1916:
1908:
1905:
1904:
1888:
1885:
1884:
1867:
1863:
1858:
1855:
1854:
1837:
1833:
1828:
1825:
1824:
1804:
1800:
1798:
1795:
1794:
1778:
1775:
1774:
1758:
1755:
1754:
1737:
1733:
1728:
1725:
1724:
1708:
1705:
1704:
1687:
1683:
1681:
1678:
1677:
1657:
1653:
1651:
1648:
1647:
1627:
1623:
1614:
1610:
1601:
1597:
1592:
1589:
1588:
1571:
1567:
1555:
1551:
1545:
1541:
1533:
1530:
1529:
1512:
1508:
1503:
1500:
1499:
1479:
1475:
1473:
1470:
1469:
1443:
1439:
1429:
1415:
1411:
1410:
1406:
1405:
1403:
1400:
1399:
1377:
1373:
1371:
1368:
1367:
1365:
1344:
1340:
1327:
1316:
1312:
1308:
1304:
1303:
1301:
1298:
1297:
1274:
1270:
1265:
1262:
1261:
1259:
1238:
1234:
1224:
1220:
1211:
1207:
1205:
1202:
1201:
1178:
1174:
1172:
1169:
1168:
1166:
1135:
1131:
1126:
1123:
1122:
1121:to themselves:
1118:
1099:
1095:
1090:
1087:
1086:
1084:
1056:
1051:
1050:
1041:
1037:
1035:
1032:
1031:
1029:
1025:
1002:
997:
996:
988:
985:
984:
981:
950:
945:
944:
935:
931:
929:
926:
925:
923:
920:identity matrix
896:
893:
892:
890:
872:
869:
868:
866:
845:
842:
841:
839:
805:
802:
801:
799:
781:
778:
777:
775:
757:
754:
753:
751:
742:") denotes the
738:(standing for "
709:
706:
705:
703:
678:
673:
672:
664:
661:
660:
658:
638:
634:
625:
621:
619:
616:
615:
599:
591:
589:
586:
585:
566:
562:
560:
557:
556:
554:
531:
527:
525:
522:
521:
519:
494:
489:
488:
480:
477:
476:
474:
446:
441:
440:
438:
435:
434:
432:
414:
412:
409:
408:
406:
382:
379:
378:
376:
358:
356:
353:
352:
350:
332:
330:
327:
326:
324:
302:
300:
297:
296:
294:
288:
282:to the kernel.
261:
258:
257:
255:
237:
234:
233:
231:
205:
201:
199:
196:
195:
193:
171:
168:
167:
165:
128:inverse element
102:Arne Bjerhammar
71:
68:
67:
65:
43:
39:
37:
34:
33:
31:
17:
12:
11:
5:
21074:
21064:
21063:
21058:
21053:
21036:
21035:
21033:
21032:
21027:
21022:
21016:
21010:
21004:
21001:Oliver Penrose
20998:
20995:Lionel Penrose
20991:
20989:
20985:
20984:
20982:
20981:
20976:
20971:
20966:
20961:
20952:
20947:
20942:
20937:
20935:Penrose stairs
20932:
20927:
20925:Penrose tiling
20922:
20917:
20912:
20907:
20902:
20897:
20892:
20887:
20882:
20877:
20872:
20867:
20862:
20857:
20852:
20850:Twistor theory
20846:
20844:
20840:
20839:
20837:
20836:
20824:
20806:
20791:
20789:
20785:
20784:
20782:
20781:
20773:
20769:Cycles of Time
20765:
20757:
20749:
20740:
20738:
20734:
20733:
20726:
20725:
20718:
20711:
20703:
20694:
20693:
20691:
20690:
20685:
20680:
20675:
20670:
20665:
20659:
20657:
20653:
20652:
20650:
20649:
20644:
20639:
20634:
20629:
20623:
20621:
20617:
20616:
20614:
20613:
20608:
20603:
20593:
20588:
20582:
20580:
20576:
20575:
20573:
20572:
20567:
20565:Floating point
20561:
20559:
20555:
20554:
20547:
20546:
20539:
20532:
20524:
20518:
20517:
20512:
20507:
20488:
20469:
20459:
20454:
20442:
20441:External links
20439:
20438:
20437:
20431:
20410:
20405:978-0201151985
20404:
20391:
20385:
20368:
20362:
20354:10.1007/b97366
20332:
20329:
20327:
20326:
20305:(4): 571–587.
20285:
20267:
20229:
20183:
20138:
20124:
20110:
20083:
20072:(4): 597–602.
20056:
20038:
20015:
19972:
19926:
19915:
19871:
19850:(5): 446–456.
19829:
19814:
19791:
19784:
19766:
19723:
19697:
19668:(3): 109–117.
19652:
19625:(4): 518–521.
19605:
19597:
19570:
19558:
19551:
19527:Golub, Gene H.
19509:
19472:Penrose, Roger
19460:
19438:
19400:
19397:
19396:
19390:
19384:
19378:
19371:
19363:
19361:
19358:
19357:
19356:
19351:
19346:
19341:
19336:
19331:
19329:Drazin inverse
19324:
19321:
19297:
19294:
19290:
19286:
19281:
19277:
19272:
19268:
19265:
19245:
19242:
19238:
19232:
19228:
19224:
19220:
19216:
19213:
19191:
19185:
19179:
19176:
19174:
19171:
19170:
19168:
19162:
19159:
19131:
19109:
19105:
19083:
19077:
19073:
19069:
19065:
19061:
19058:
19055:
19051:
19047:
19042:
19038:
19033:
19029:
19026:
19023:
19020:
19017:
19014:
19011:
19008:
18970:
18966:
18935:
18931:
18904:
18900:
18887:Hilbert spaces
18870:
18866:
18862:
18857:
18853:
18849:
18846:
18832:
18829:
18814:
18790:
18786:
18781:
18777:
18773:
18769:
18766:
18763:
18760:
18757:
18754:
18751:
18724:
18721:
18720:
18719:
18704:
18699:
18695:
18692:
18689:
18683:
18678:
18674:
18671:
18651:
18646:
18642:
18638:
18635:
18615:
18612:
18609:
18606:
18578:
18575:
18572:
18567:
18562:
18559:
18546:
18545:
18531:
18527:
18523:
18520:
18517:
18512:
18508:
18504:
18501:
18481:
18476:
18472:
18468:
18465:
18445:
18442:
18439:
18436:
18410:
18406:
18402:
18399:
18389:Euclidean norm
18376:
18373:
18370:
18367:
18364:
18352:
18349:
18330:
18306:
18301:
18297:
18293:
18290:
18266:
18244:
18241:
18238:
18233:
18229:
18225:
18203:
18179:
18175:
18171:
18166:
18162:
18158:
18155:
18151:
18147:
18144:
18139:
18135:
18131:
18128:
18104:
18101:
18098:
18095:
18081:
18078:
18077:
18076:
18073:Frobenius norm
18057:
18052:
18048:
18045:
18025:
18020:
18016:
18012:
18009:
17986:
17981:
17977:
17974:
17971:
17968:
17965:
17962:
17956:
17951:
17947:
17944:
17941:
17938:
17935:
17911:
17908:
17905:
17900:
17895:
17892:
17889:
17860:
17857:
17854:
17849:
17844:
17841:
17828:
17827:
17814:
17811:
17786:
17782:
17777:
17773:
17769:
17766:
17762:
17737:
17713:
17691:
17687:
17683:
17678:
17674:
17670:
17667:
17663:
17659:
17656:
17651:
17647:
17643:
17640:
17629:Euclidean norm
17614:
17610:
17606:
17603:
17583:
17578:
17574:
17570:
17567:
17545:
17540:
17536:
17533:
17530:
17527:
17523:
17518:
17513:
17508:
17504:
17501:
17498:
17495:
17491:
17465:
17460:
17455:
17452:
17449:
17424:
17404:
17384:
17362:
17338:
17335:
17332:
17329:
17326:
17302:
17299:
17296:
17291:
17286:
17283:
17268:solution to a
17255:
17252:
17250:
17247:
17207:function. The
17150:
17147:
17123:
17099:
17094:
17090:
17065:
17039:
17035:
17031:
17007:
16993:
16990:
16975:
16971:
16965:
16962:
16957:
16953:
16950:
16947:
16944:
16939:
16935:
16930:
16925:
16920:
16916:
16906:, for example
16893:
16888:
16884:
16879:
16875:
16870:
16865:
16862:
16857:
16853:
16828:
16824:
16797:
16793:
16768:
16744:
16741:
16738:
16733:
16729:
16706:
16703:
16700:
16695:
16690:
16686:
16681:
16677:
16674:
16671:
16668:
16665:
16643:
16639:
16635:
16632:
16627:
16623:
16600:
16595:
16591:
16586:
16582:
16577:
16572:
16569:
16564:
16560:
16535:
16531:
16506:
16482:
16477:
16473:
16469:
16464:
16460:
16456:
16451:
16447:
16443:
16440:
16435:
16432:
16429:
16425:
16413:Drazin inverse
16408:
16405:
16397:
16396:Block matrices
16394:
16379:
16355:
16331:
16265:
16237:
16233:
16227:
16223:
16219:
16216:
16211:
16207:
16184:
16160:
16156:
16152:
16149:
16146:
16143:
16127:
16124:
16112:
16109:
16104:
16100:
16096:
16093:
16090:
16085:
16081:
16077:
16072:
16068:
16064:
16061:
16056:
16052:
16048:
16045:
16042:
16037:
16033:
16010:
15988:
15985:
15980:
15976:
15972:
15969:
15966:
15961:
15957:
15953:
15948:
15944:
15940:
15920:
15917:
15914:
15911:
15891:
15886:
15882:
15857:
15854:
15851:
15846:
15841:
15838:
15824:
15821:
15804:
15800:
15775:
15751:
15748:
15743:
15737:
15733:
15729:
15725:
15718:
15714:
15710:
15705:
15701:
15675:
15670:
15666:
15641:
15617:
15614:
15609:
15605:
15600:
15596:
15593:
15588:
15584:
15578:
15574:
15569:
15565:
15562:
15559:
15556:
15551:
15547:
15543:
15540:
15537:
15533:
15529:
15524:
15520:
15497:
15475:
15472:
15469:
15464:
15460:
15439:
15419:
15416:
15413:
15410:
15384:
15379:
15375:
15337:
15333:
15329:
15324:
15320:
15316:
15311:
15307:
15302:
15296:
15292:
15288:
15283:
15279:
15274:
15270:
15265:
15261:
15256:
15251:
15245:
15241:
15235:
15232:
15227:
15223:
15218:
15214:
15209:
15204:
15199:
15195:
15168:
15146:
15141:
15137:
15133:
15130:
15125:
15121:
15095:
15091:
15085:
15082:
15077:
15073:
15068:
15064:
15059:
15054:
15049:
15045:
15022:
14995:
14967:
14962:
14958:
14931:
14927:
14923:
14901:
14897:
14893:
14889:
14885:
14882:
14878:
14865:
14862:
14847:
14843:
14837:
14834:
14829:
14825:
14820:
14816:
14811:
14804:
14801:
14796:
14790:
14786:
14782:
14778:
14771:
14767:
14763:
14758:
14754:
14748:
14744:
14740:
14735:
14731:
14708:
14682:
14679:
14676:
14671:
14666:
14663:
14637:
14634:
14631:
14626:
14621:
14618:
14596:
14593:
14590:
14587:
14561:
14535:
14532:
14529:
14524:
14519:
14516:
14488:
14485:
14482:
14479:
14476:
14473:
14470:
14467:
14453:
14450:
14448:
14445:
14429:
14424:
14418:
14412:
14407:
14403:
14399:
14394:
14389:
14386:
14384:
14380:
14376:
14372:
14371:
14368:
14363:
14357:
14351:
14348:
14345:
14340:
14335:
14332:
14330:
14328:
14325:
14324:
14296:
14266:
14249:
14246:
14234:
14231:
14228:
14223:
14219:
14198:
14195:
14190:
14186:
14163:
14159:
14155:
14152:
14130:
14116:
14113:
14098:
14074:
14052:
14049:
14046:
14043:
14040:
14035:
14031:
14010:
14007:
14004:
14001:
13979:
13955:
13941:
13938:
13923:
13895:
13881:
13878:
13866:
13861:
13857:
13853:
13848:
13844:
13821:
13817:
13813:
13808:
13804:
13800:
13778:
13774:
13770:
13767:
13762:
13758:
13735:
13721:
13718:
13703:
13699:
13695:
13690:
13686:
13682:
13660:
13634:
13630:
13605:
13600:
13597:
13592:
13586:
13582:
13578:
13574:
13567:
13563:
13559:
13554:
13550:
13525:
13521:
13517:
13490:
13466:
13463:
13460:
13436:
13412:
13398:
13395:
13380:
13376:
13372:
13369:
13364:
13360:
13337:
13311:
13307:
13282:
13277:
13273:
13267:
13264:
13259:
13255:
13250:
13246:
13241:
13236:
13231:
13227:
13204:
13199:
13195:
13167:
13143:
13140:
13137:
13113:
13089:
13075:
13072:
13057:
13054:
13050:
13045:
13041:
13038:
13033:
13028:
13025:
13021:
13016:
13012:
13009:
13004:
13001:
12997:
12976:
12973:
12970:
12948:
12944:
12922:
12918:
12913:
12909:
12886:
12882:
12876:
12873:
12869:
12865:
12860:
12857:
12853:
12832:
12829:
12826:
12806:
12784:
12781:
12778:
12773:
12768:
12763:
12760:
12753:
12750:
12743:
12738:
12734:
12713:
12710:
12704:
12701:
12676:
12673:
12670:
12665:
12660:
12654:
12651:
12645:
12642:
12622:
12610:
12607:
12593:
12588:
12580:
12578:
12575:
12569:
12566:
12558:
12551:
12548:
12541:
12537:
12531:
12524:
12521:
12514:
12513:
12510:
12504:
12501:
12495:
12489:
12486:
12475:
12473:
12468:
12461:
12458:
12451:
12450:
12448:
12443:
12438:
12431:
12428:
12410:
12407:
12393:
12388:
12378:
12376:
12371:
12368:
12364:
12360:
12359:
12356:
12353:
12350:
12347:
12337:
12335:
12332:
12329:
12328:
12326:
12321:
12316:
12312:
12289:
12265:
12241:
12227:
12224:
12222:
12219:
12215:
12214:
12203:
12198:
12192:
12189:
12187:
12184:
12183:
12180:
12177:
12175:
12172:
12171:
12169:
12164:
12161:
12156:
12152:
12129:
12125:
12108:
12107:
12093:
12085:
12082:
12077:
12073:
12070:
12065:
12063:
12060:
12059:
12056:
12053:
12051:
12048:
12046:
12043:
12042:
12040:
12035:
12030:
12026:
12005:
12000:
11994:
11991:
11989:
11986:
11985:
11982:
11979:
11977:
11974:
11973:
11970:
11967:
11965:
11962:
11961:
11959:
11954:
11951:
11940:
11929:
11924:
11916:
11913:
11908:
11904:
11901:
11896:
11895:
11890:
11887:
11882:
11878:
11875:
11870:
11869:
11867:
11862:
11857:
11853:
11833:
11828:
11822:
11819:
11817:
11814:
11813:
11810:
11807:
11805:
11802:
11801:
11799:
11794:
11791:
11780:
11766:
11762:
11758:
11753:
11749:
11745:
11742:
11720:
11714:
11711:
11709:
11706:
11705:
11700:
11697:
11692:
11688:
11685:
11680:
11679:
11677:
11672:
11667:
11663:
11643:
11638:
11632:
11629:
11627:
11624:
11623:
11620:
11617:
11615:
11612:
11611:
11609:
11604:
11601:
11590:
11579:
11574:
11568:
11565:
11563:
11560:
11559:
11554:
11551:
11546:
11543:
11539:
11536:
11531:
11530:
11528:
11523:
11518:
11514:
11494:
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11483:
11480:
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11475:
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11468:
11465:
11463:
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11457:
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11437:
11436:
11422:
11417:
11411:
11404:
11399:
11394:
11389:
11384:
11376:
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11364:
11355:
11351:
11346:
11341:
11337:
11314:
11309:
11304:
11298:
11292:
11285:
11280:
11275:
11270:
11265:
11257:
11251:
11246:
11243:
11232:
11218:
11214:
11191:
11187:
11183:
11163:
11158:
11154:
11131:
11128:
11123:
11117:
11113:
11109:
11105:
11098:
11094:
11090:
11085:
11081:
11058:
11054:
11048:
11045:
11040:
11036:
11031:
11027:
11022:
11017:
11012:
11008:
10985:
10972:
10958:
10954:
10950:
10945:
10939:
10936:
10934:
10931:
10930:
10925:
10922:
10917:
10913:
10910:
10905:
10904:
10902:
10897:
10892:
10888:
10883:
10878:
10874:
10851:
10845:
10842:
10840:
10837:
10836:
10833:
10830:
10828:
10825:
10824:
10822:
10817:
10814:
10809:
10805:
10784:
10781:
10776:
10770:
10767:
10765:
10762:
10761:
10758:
10755:
10753:
10750:
10749:
10747:
10742:
10739:
10734:
10730:
10725:
10703:
10695:
10692:
10687:
10683:
10680:
10675:
10674:
10669:
10666:
10661:
10657:
10654:
10649:
10648:
10646:
10641:
10636:
10632:
10627:
10615:
10614:
10600:
10594:
10591:
10589:
10586:
10585:
10580:
10577:
10572:
10568:
10565:
10560:
10559:
10557:
10552:
10547:
10543:
10522:
10517:
10511:
10508:
10506:
10503:
10502:
10499:
10496:
10494:
10491:
10490:
10488:
10483:
10480:
10469:
10455:
10451:
10447:
10442:
10438:
10434:
10429:
10425:
10404:
10399:
10393:
10390:
10388:
10385:
10384:
10381:
10378:
10376:
10373:
10372:
10370:
10365:
10360:
10356:
10335:
10330:
10324:
10321:
10319:
10316:
10315:
10312:
10309:
10307:
10304:
10303:
10301:
10296:
10293:
10277:
10274:
10257:
10252:
10248:
10244:
10239:
10235:
10231:
10226:
10222:
10201:
10196:
10192:
10188:
10185:
10182:
10179:
10157:
10153:
10130:
10126:
10105:
10085:
10080:
10076:
10070:
10067:
10063:
10058:
10052:
10048:
10040:
10036:
10031:
10024:
10019:
10015:
10010:
10006:
10002:
9999:
9995:
9988:
9981:
9975:
9971:
9967:
9964:
9961:
9957:
9952:
9946:
9942:
9934:
9930:
9925:
9918:
9912:
9909:
9905:
9899:
9895:
9888:
9880:
9876:
9871:
9865:
9861:
9855:
9851:
9844:
9838:
9834:
9830:
9827:
9822:
9818:
9803:
9799:
9795:
9792:
9787:
9781:
9777:
9771:
9765:
9742:
9738:
9717:
9695:
9691:
9670:
9667:
9664:
9659:
9655:
9651:
9648:
9624:
9620:
9597:
9594:
9591:
9588:
9585:
9582:
9570:
9567:
9550:
9546:
9519:
9515:
9509:
9505:
9501:
9477:
9451:
9447:
9420:
9416:
9389:
9385:
9379:
9375:
9371:
9343:
9318:
9313:
9309:
9305:
9286:
9283:
9281:do not exist.
9266:
9263:
9258:
9254:
9249:
9245:
9240:
9212:
9209:
9204:
9198:
9194:
9190:
9186:
9156:
9153:
9148:
9144:
9141:
9138:
9133:
9129:
9125:
9121:
9114:
9110:
9104:
9101:
9098:
9094:
9090:
9085:
9081:
9075:
9072:
9067:
9063:
9060:
9057:
9054:
9049:
9045:
9040:
9033:
9030:
9027:
9023:
9019:
9014:
9010:
8997:
8994:
8972:
8948:
8945:
8942:
8939:
8936:
8933:
8930:
8927:
8922:
8919:
8915:
8890:
8885:
8881:
8856:
8830:
8825:
8799:
8796:
8793:
8790:
8766:
8740:
8735:
8709:
8706:
8703:
8700:
8697:
8694:
8691:
8688:
8683:
8680:
8676:
8651:
8648:
8645:
8642:
8618:
8594:
8568:
8563:
8537:
8513:
8508:
8504:
8479:
8474:
8469:
8465:
8462:
8459:
8455:
8431:
8428:
8425:
8399:
8395:
8372:
8369:
8366:
8363:
8358:
8353:
8349:
8346:
8343:
8339:
8334:
8331:
8309:
8304:
8300:
8297:
8294:
8290:
8285:
8282:
8279:
8276:
8273:
8268:
8263:
8241:
8238:
8235:
8232:
8227:
8222:
8218:
8215:
8212:
8208:
8203:
8198:
8193:
8169:
8158:of a map, and
8141:
8113:
8085:
8059:
8054:
8049:
8044:
8039:
8034:
8029:
8025:
7999:
7972:
7967:
7962:
7957:
7952:
7947:
7944:
7930:
7927:
7914:
7911:
7906:
7902:
7879:
7853:
7849:
7845:
7842:
7839:
7836:
7833:
7828:
7824:
7820:
7817:
7814:
7791:
7788:
7785:
7780:
7775:
7772:
7746:
7743:
7740:
7735:
7730:
7727:
7700:
7676:
7652:
7628:
7624:
7620:
7617:
7614:
7611:
7608:
7605:
7585:
7582:
7579:
7576:
7552:
7548:
7544:
7541:
7538:
7535:
7530:
7526:
7522:
7519:
7516:
7513:
7489:
7486:
7483:
7478:
7473:
7470:
7444:
7441:
7438:
7433:
7428:
7425:
7412:
7411:
7400:
7397:
7392:
7388:
7383:
7379:
7374:
7370:
7366:
7363:
7359:
7355:
7351:
7345:
7341:
7337:
7334:
7331:
7327:
7321:
7317:
7306:
7295:
7292:
7283:
7279:
7273:
7269:
7265:
7262:
7259:
7255:
7251:
7247:
7243:
7238:
7234:
7230:
7227:
7223:
7215:
7201:
7200:
7184:
7180:
7155:
7151:
7147:
7144:
7141:
7138:
7135:
7132:
7129:
7119:
7105:
7083:
7078:
7074:
7070:
7067:
7064:
7061:
7058:
7055:
7045:
7031:
7005:
7001:
6976:
6964:
6948:
6944:
6915:
6883:
6871:
6858:
6854:
6850:
6845:
6841:
6837:
6834:
6831:
6826:
6822:
6801:
6798:
6795:
6792:
6789:
6786:
6783:
6760:
6757:
6752:
6748:
6727:
6724:
6719:
6715:
6692:
6688:
6684:
6681:
6659:
6655:
6651:
6648:
6624:
6619:
6615:
6611:
6608:
6586:
6582:
6578:
6575:
6572:
6561:
6558:
6544:
6538:
6532:
6529:
6527:
6524:
6523:
6520:
6517:
6515:
6512:
6511:
6509:
6501:
6495:
6489:
6486:
6484:
6481:
6480:
6477:
6474:
6472:
6469:
6468:
6466:
6460:
6455:
6449:
6446:
6441:
6438:
6432:
6431:
6428:
6425:
6420:
6417:
6411:
6410:
6408:
6401:
6392:
6389:
6383:
6381:
6378:
6377:
6371:
6368:
6362:
6360:
6357:
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6349:
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6338:
6335:
6330:
6327:
6321:
6320:
6317:
6314:
6309:
6306:
6300:
6299:
6297:
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6271:
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6262:
6261:
6258:
6255:
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6241:
6240:
6238:
6233:
6228:
6222:
6216:
6213:
6211:
6208:
6207:
6204:
6201:
6199:
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6195:
6193:
6187:
6182:
6176:
6168:
6162:
6159:
6157:
6154:
6153:
6150:
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6145:
6142:
6141:
6139:
6132:
6126:
6123:
6121:
6118:
6117:
6114:
6111:
6109:
6106:
6105:
6103:
6096:
6068:
6064:
6058:
6054:
6050:
6045:
6041:
6037:
6034:
6031:
6003:
5998:
5995:
5991:
5985:
5981:
5977:
5974:
5972:
5968:
5963:
5959:
5954:
5950:
5945:
5940:
5939:
5936:
5931:
5927:
5921:
5918:
5914:
5910:
5907:
5905:
5901:
5896:
5890:
5886:
5882:
5878:
5873:
5872:
5861:
5860:
5849:
5846:
5841:
5837:
5833:
5830:
5825:
5821:
5817:
5814:
5811:
5808:
5803:
5799:
5795:
5792:
5789:
5786:
5781:
5777:
5773:
5746:
5742:
5736:
5732:
5728:
5723:
5719:
5715:
5712:
5709:
5696:
5695:
5681:
5677:
5673:
5670:
5660:
5646:
5642:
5638:
5635:
5625:
5613:
5610:
5605:
5601:
5597:
5575:
5553:
5550:
5547:
5542:
5538:
5515:
5503:
5489:
5485:
5481:
5476:
5472:
5468:
5465:
5460:
5456:
5452:
5430:
5418:
5404:
5400:
5396:
5393:
5388:
5384:
5380:
5377:
5372:
5368:
5345:
5316:
5312:
5306:
5302:
5298:
5293:
5289:
5285:
5282:
5279:
5266:
5265:
5250:
5245:
5241:
5237:
5234:
5231:
5228:
5225:
5220:
5216:
5212:
5209:
5207:
5203:
5199:
5193:
5189:
5185:
5182:
5181:
5178:
5175:
5172:
5167:
5163:
5159:
5156:
5153:
5150:
5147:
5144:
5142:
5140:
5137:
5132:
5128:
5124:
5123:
5113:
5102:
5097:
5093:
5087:
5083:
5079:
5076:
5071:
5067:
5063:
5060:
5055:
5051:
5045:
5041:
5037:
5034:
5029:
5025:
5014:
4999:
4994:
4990:
4986:
4983:
4978:
4974:
4970:
4967:
4965:
4961:
4956:
4950:
4946:
4942:
4939:
4934:
4930:
4925:
4920:
4919:
4916:
4911:
4907:
4903:
4900:
4895:
4891:
4887:
4884:
4882:
4878:
4873:
4867:
4863:
4859:
4856:
4851:
4847:
4842:
4837:
4836:
4826:
4811:
4808:
4805:
4800:
4796:
4792:
4789:
4787:
4785:
4782:
4777:
4773:
4767:
4763:
4759:
4756:
4755:
4752:
4747:
4743:
4737:
4733:
4729:
4726:
4723:
4721:
4717:
4713:
4707:
4703:
4699:
4696:
4691:
4687:
4683:
4682:
4672:
4659:
4655:
4649:
4645:
4641:
4636:
4632:
4628:
4625:
4622:
4595:
4592:
4589:
4584:
4579:
4576:
4570:
4565:
4562:
4559:
4554:
4549:
4546:
4520:
4516:
4510:
4506:
4502:
4497:
4493:
4489:
4486:
4483:
4469:
4466:
4449:
4445:
4441:
4417:
4412:
4408:
4383:
4378:
4373:
4367:
4363:
4359:
4355:
4348:
4344:
4340:
4335:
4331:
4311:
4306:
4302:
4296:
4291:
4287:
4282:
4278:
4273:
4268:
4263:
4259:
4246:
4243:
4231:
4226:
4222:
4216:
4209:
4205:
4199:
4192:
4188:
4182:
4175:
4171:
4165:
4160:
4156:
4135:
4113:
4109:
4088:
4083:
4079:
4071:
4068:
4064:
4056:
4052:
4046:
4039:
4035:
4027:
4024:
4020:
4012:
4008:
4002:
3997:
3993:
3972:
3950:
3946:
3925:
3922:
3915:
3911:
3903:
3900:
3896:
3890:
3883:
3880:
3876:
3868:
3864:
3858:
3853:
3850:
3838:
3835:
3834:
3833:
3820:
3815:
3811:
3807:
3803:
3800:
3797:
3793:
3788:
3784:
3780:
3776:
3773:
3752:
3747:
3743:
3739:
3735:
3732:
3729:
3725:
3720:
3716:
3712:
3708:
3705:
3694:
3680:
3677:
3674:
3650:
3646:
3640:
3637:
3633:
3629:
3624:
3619:
3615:
3612:
3608:
3582:
3578:
3553:
3540:
3529:
3524:
3519:
3514:
3510:
3506:
3501:
3496:
3491:
3486:
3482:
3478:
3472:
3467:
3462:
3458:
3452:
3447:
3442:
3437:
3434:
3429:
3423:
3418:
3413:
3408:
3404:
3400:
3395:
3390:
3385:
3380:
3376:
3372:
3359:
3347:
3344:
3339:
3334:
3329:
3325:
3321:
3308:
3294:
3291:
3287:
3283:
3278:
3274:
3247:
3234:
3218:
3214:
3189:
3167:
3164:
3141:
3137:
3133:
3128:
3124:
3120:
3115:
3111:
3090:
3087:
3084:
3079:
3075:
3071:
3044:
3040:
3015:
3001:
2998:
2996:
2993:
2980:
2958:
2954:
2950:
2947:
2944:
2939:
2935:
2912:
2908:
2904:
2901:
2896:
2892:
2888:
2868:
2846:
2842:
2819:
2815:
2809:
2805:
2801:
2798:
2793:
2789:
2768:
2748:
2745:
2742:
2720:
2716:
2712:
2709:
2706:
2703:
2688:
2687:
2675:
2672:
2667:
2663:
2659:
2635:
2630:
2627:
2622:
2616:
2612:
2608:
2604:
2597:
2593:
2589:
2584:
2580:
2555:
2551:
2524:
2520:
2516:
2494:
2474:
2463:
2451:
2448:
2445:
2440:
2436:
2411:
2406:
2402:
2396:
2393:
2388:
2384:
2379:
2375:
2370:
2365:
2360:
2356:
2331:
2327:
2302:
2297:
2293:
2268:
2244:
2214:
2210:
2185:
2182:
2179:
2176:
2173:
2170:
2144:
2122:
2119:
2116:
2111:
2106:
2103:
2081:
2077:
2051:
2047:
2043:
2038:
2034:
2030:
2027:
2022:
2018:
2014:
1992:
1988:
1967:
1962:
1958:
1937:
1934:
1931:
1928:
1923:
1919:
1915:
1912:
1892:
1870:
1866:
1862:
1840:
1836:
1832:
1812:
1807:
1803:
1782:
1762:
1740:
1736:
1732:
1712:
1690:
1686:
1665:
1660:
1656:
1635:
1630:
1626:
1622:
1617:
1613:
1609:
1604:
1600:
1596:
1574:
1570:
1566:
1563:
1558:
1554:
1548:
1544:
1540:
1537:
1515:
1511:
1507:
1487:
1482:
1478:
1466:
1465:
1454:
1451:
1446:
1442:
1437:
1432:
1427:
1423:
1418:
1414:
1409:
1385:
1380:
1376:
1363:
1352:
1347:
1343:
1339:
1335:
1330:
1325:
1319:
1315:
1311:
1307:
1277:
1273:
1269:
1257:
1246:
1241:
1237:
1232:
1227:
1223:
1219:
1214:
1210:
1181:
1177:
1164:
1153:
1150:
1146:
1143:
1138:
1134:
1130:
1102:
1098:
1094:
1065:
1062:
1059:
1054:
1049:
1044:
1040:
1011:
1008:
1005:
1000:
995:
992:
980:
977:
976:
975:
959:
956:
953:
948:
943:
938:
934:
906:
903:
900:
876:
863:
849:
821:
818:
815:
812:
809:
785:
761:
725:
722:
719:
716:
713:
687:
684:
681:
676:
671:
668:
655:
641:
637:
633:
628:
624:
602:
598:
594:
569:
565:
534:
530:
503:
500:
497:
492:
487:
484:
471:
455:
452:
449:
444:
431:is denoted by
417:
405:matrices over
392:
389:
386:
361:
335:
305:
287:
284:
265:
241:
208:
204:
175:
143:§ Applications
94:inverse matrix
75:
46:
42:
25:linear algebra
15:
9:
6:
4:
3:
2:
21073:
21062:
21059:
21057:
21054:
21052:
21051:Matrix theory
21049:
21048:
21046:
21031:
21028:
21026:
21023:
21021:(grandfather)
21020:
21017:
21014:
21011:
21008:
21005:
21002:
20999:
20996:
20993:
20992:
20990:
20986:
20980:
20977:
20975:
20972:
20970:
20967:
20965:
20962:
20960:
20956:
20953:
20951:
20948:
20946:
20943:
20941:
20938:
20936:
20933:
20931:
20928:
20926:
20923:
20921:
20918:
20916:
20913:
20911:
20908:
20906:
20903:
20901:
20898:
20896:
20893:
20891:
20888:
20886:
20883:
20881:
20878:
20876:
20873:
20871:
20868:
20866:
20863:
20861:
20858:
20856:
20853:
20851:
20848:
20847:
20845:
20841:
20833:
20829:
20825:
20821:
20817:
20816:Abner Shimony
20813:
20812:
20807:
20803:
20799:
20798:
20793:
20792:
20790:
20786:
20779:
20778:
20774:
20771:
20770:
20766:
20763:
20762:
20758:
20755:
20754:
20750:
20747:
20746:
20742:
20741:
20739:
20735:
20731:
20730:Roger Penrose
20724:
20719:
20717:
20712:
20710:
20705:
20704:
20701:
20689:
20686:
20684:
20681:
20679:
20676:
20674:
20671:
20669:
20666:
20664:
20661:
20660:
20658:
20654:
20648:
20645:
20643:
20640:
20638:
20635:
20633:
20630:
20628:
20625:
20624:
20622:
20618:
20612:
20609:
20607:
20604:
20601:
20597:
20594:
20592:
20589:
20587:
20584:
20583:
20581:
20577:
20571:
20568:
20566:
20563:
20562:
20560:
20556:
20552:
20545:
20540:
20538:
20533:
20531:
20526:
20525:
20522:
20516:
20513:
20511:
20508:
20503:
20502:
20497:
20494:
20489:
20484:
20483:
20478:
20475:
20470:
20467:
20463:
20460:
20458:
20455:
20452:
20448:
20447:Pseudoinverse
20445:
20444:
20434:
20428:
20424:
20419:
20418:
20411:
20407:
20401:
20397:
20392:
20388:
20382:
20377:
20376:
20369:
20365:
20359:
20355:
20351:
20347:
20343:
20339:
20335:
20334:
20322:
20318:
20313:
20308:
20304:
20300:
20296:
20289:
20281:
20274:
20272:
20263:
20259:
20255:
20251:
20247:
20243:
20236:
20234:
20225:
20221:
20217:
20213:
20209:
20205:
20201:
20197:
20190:
20188:
20179:
20175:
20171:
20167:
20163:
20159:
20155:
20151:
20150:
20142:
20134:
20128:
20120:
20114:
20106:
20102:
20099:(3): 315–23.
20098:
20094:
20087:
20079:
20075:
20071:
20067:
20060:
20049:
20042:
20034:
20030:
20026:
20019:
20011:
20007:
20003:
19999:
19995:
19991:
19987:
19983:
19976:
19970:
19965:
19961:
19957:
19953:
19949:
19945:
19942:(3): 410–19.
19941:
19937:
19930:
19924:
19919:
19911:
19907:
19903:
19899:
19896:(2): 385–88.
19895:
19891:
19890:
19885:
19881:
19875:
19866:
19861:
19857:
19853:
19849:
19845:
19844:
19836:
19834:
19826:
19821:
19819:
19810:
19806:
19802:
19795:
19787:
19785:9780521192644
19781:
19777:
19770:
19762:
19758:
19754:
19750:
19746:
19742:
19739:(2): 413–32.
19738:
19734:
19727:
19719:
19715:
19708:
19701:
19693:
19689:
19684:
19679:
19675:
19671:
19667:
19663:
19656:
19648:
19644:
19640:
19636:
19632:
19628:
19624:
19620:
19616:
19609:
19600:
19594:
19590:
19586:
19579:
19577:
19575:
19567:
19562:
19554:
19548:
19544:
19539:
19538:
19532:
19528:
19522:
19520:
19518:
19516:
19514:
19504:
19499:
19495:
19491:
19488:(3): 406–13.
19487:
19483:
19482:
19477:
19473:
19467:
19465:
19456:
19452:
19448:
19442:
19433:
19428:
19425:(9): 394–95.
19424:
19420:
19419:
19414:
19410:
19404:
19394:
19391:
19388:
19387:Nakamura 1991
19385:
19382:
19379:
19376:
19373:
19372:
19368:
19364:
19355:
19352:
19350:
19347:
19345:
19342:
19340:
19337:
19335:
19332:
19330:
19327:
19326:
19320:
19318:
19314:
19309:
19295:
19292:
19288:
19284:
19275:
19270:
19266:
19263:
19243:
19240:
19236:
19226:
19222:
19218:
19214:
19211:
19183:
19177:
19172:
19166:
19160:
19157:
19149:
19145:
19129:
19107:
19103:
19081:
19075:
19071:
19067:
19063:
19059:
19056:
19053:
19049:
19045:
19040:
19036:
19031:
19027:
19024:
19021:
19015:
19009:
19006:
18998:
18995:
18991:
18986:
18968:
18964:
18953:
18933:
18929:
18902:
18898:
18888:
18868:
18864:
18855:
18851:
18847:
18844:
18828:
18812:
18801:
18788:
18779:
18775:
18764:
18758:
18752:
18734:
18730:
18693:
18687:
18672:
18649:
18644:
18640:
18636:
18633:
18613:
18610:
18607:
18604:
18596:
18595:
18594:
18576:
18573:
18570:
18560:
18557:
18529:
18521:
18515:
18510:
18502:
18479:
18474:
18470:
18466:
18463:
18443:
18440:
18437:
18434:
18426:
18425:
18424:
18408:
18400:
18390:
18374:
18371:
18368:
18365:
18362:
18348:
18346:
18328:
18304:
18299:
18295:
18291:
18288:
18264:
18242:
18239:
18236:
18231:
18227:
18223:
18201:
18190:
18177:
18173:
18169:
18164:
18160:
18156:
18153:
18149:
18145:
18142:
18137:
18133:
18129:
18126:
18118:
18115:
18102:
18099:
18096:
18093:
18085:
18074:
18046:
18023:
18018:
18014:
18010:
18007:
17975:
17972:
17969:
17966:
17960:
17945:
17942:
17939:
17936:
17909:
17906:
17903:
17893:
17890:
17878:
17877:
17876:
17858:
17855:
17852:
17842:
17839:
17812:
17809:
17784:
17780:
17775:
17771:
17767:
17764:
17760:
17735:
17711:
17689:
17685:
17681:
17676:
17672:
17668:
17665:
17661:
17657:
17654:
17649:
17645:
17641:
17638:
17630:
17612:
17604:
17581:
17576:
17572:
17568:
17565:
17543:
17534:
17531:
17528:
17525:
17516:
17511:
17502:
17499:
17496:
17493:
17463:
17453:
17450:
17438:
17437:
17436:
17422:
17402:
17382:
17360:
17349:
17336:
17333:
17330:
17327:
17324:
17300:
17297:
17294:
17284:
17281:
17271:
17267:
17266:least squares
17261:
17246:
17240:
17235:
17225:
17220:
17202:
17197:
17191:
17175:
17170:
17168:
17164:
17160:
17156:
17146:
17142:
17139:
17121:
17097:
17092:
17088:
17063:
17037:
17033:
17029:
17005:
16989:
16973:
16969:
16963:
16960:
16955:
16951:
16948:
16945:
16942:
16937:
16933:
16928:
16923:
16918:
16914:
16891:
16886:
16882:
16877:
16873:
16868:
16863:
16860:
16855:
16851:
16826:
16822:
16795:
16791:
16766:
16739:
16731:
16727:
16701:
16693:
16688:
16684:
16679:
16675:
16672:
16669:
16666:
16663:
16641:
16637:
16633:
16630:
16625:
16621:
16613:. The choice
16598:
16593:
16589:
16584:
16580:
16575:
16570:
16567:
16562:
16558:
16533:
16529:
16504:
16493:
16480:
16475:
16471:
16467:
16462:
16458:
16454:
16449:
16445:
16441:
16438:
16433:
16430:
16427:
16423:
16414:
16404:
16402:
16393:
16377:
16329:
16318:
16316:
16310:
16308:
16302:
16298:
16294:
16285:
16281:
16253:
16235:
16231:
16225:
16217:
16214:
16209:
16205:
16182:
16158:
16154:
16147:
16144:
16141:
16133:
16123:
16110:
16102:
16098:
16094:
16088:
16083:
16079:
16075:
16070:
16066:
16059:
16054:
16050:
16043:
16040:
16035:
16031:
16008:
15983:
15978:
15974:
15967:
15964:
15959:
15951:
15946:
15942:
15915:
15909:
15889:
15884:
15880:
15855:
15852:
15849:
15839:
15836:
15820:
15802:
15798:
15773:
15749:
15746:
15741:
15735:
15731:
15727:
15723:
15716:
15712:
15708:
15703:
15699:
15689:
15673:
15668:
15664:
15639:
15628:
15615:
15612:
15607:
15603:
15598:
15594:
15591:
15586:
15582:
15576:
15572:
15567:
15560:
15557:
15549:
15541:
15538:
15531:
15527:
15522:
15518:
15495:
15473:
15470:
15467:
15462:
15458:
15437:
15417:
15414:
15411:
15408:
15400:
15382:
15377:
15373:
15361:
15359:
15355:
15350:
15335:
15331:
15327:
15322:
15318:
15314:
15309:
15305:
15294:
15290:
15286:
15281:
15277:
15272:
15268:
15263:
15259:
15254:
15243:
15239:
15233:
15230:
15225:
15221:
15216:
15212:
15207:
15202:
15197:
15193:
15184:
15166:
15144:
15139:
15135:
15131:
15128:
15123:
15119:
15111:
15093:
15089:
15083:
15080:
15075:
15071:
15066:
15062:
15057:
15052:
15047:
15043:
15020:
15009:
14993:
14983:
14965:
14960:
14956:
14929:
14925:
14921:
14891:
14880:
14864:The QR method
14861:
14845:
14841:
14835:
14832:
14827:
14823:
14818:
14814:
14809:
14802:
14799:
14794:
14788:
14784:
14780:
14776:
14769:
14765:
14761:
14756:
14752:
14746:
14742:
14738:
14733:
14729:
14706:
14680:
14677:
14674:
14664:
14661:
14635:
14632:
14629:
14619:
14616:
14594:
14591:
14588:
14585:
14577:
14559:
14533:
14530:
14527:
14517:
14514:
14504:
14483:
14480:
14477:
14468:
14465:
14444:
14427:
14422:
14410:
14405:
14397:
14387:
14385:
14378:
14374:
14366:
14361:
14349:
14343:
14333:
14331:
14326:
14314:
14282:
14264:
14255:
14245:
14232:
14229:
14226:
14221:
14217:
14196:
14193:
14188:
14184:
14161:
14157:
14153:
14150:
14128:
14112:
14096:
14072:
14047:
14041:
14038:
14033:
14029:
14005:
13999:
13977:
13953:
13937:
13921:
13911:
13893:
13877:
13864:
13859:
13855:
13851:
13846:
13842:
13819:
13815:
13811:
13806:
13802:
13798:
13776:
13772:
13768:
13765:
13760:
13756:
13733:
13717:
13701:
13697:
13693:
13688:
13684:
13680:
13658:
13632:
13628:
13616:
13603:
13598:
13595:
13590:
13584:
13580:
13576:
13572:
13565:
13561:
13557:
13552:
13548:
13523:
13519:
13515:
13505:
13488:
13479:,) there are
13464:
13461:
13458:
13434:
13410:
13394:
13378:
13374:
13370:
13367:
13362:
13358:
13335:
13309:
13305:
13293:
13280:
13275:
13271:
13265:
13262:
13257:
13253:
13248:
13244:
13239:
13234:
13229:
13225:
13202:
13197:
13193:
13183:columns, and
13182:
13165:
13156:,) there are
13141:
13138:
13135:
13111:
13087:
13071:
13055:
13052:
13048:
13043:
13039:
13036:
13031:
13026:
13023:
13019:
13010:
13007:
13002:
12999:
12995:
12974:
12971:
12968:
12946:
12942:
12916:
12911:
12907:
12884:
12880:
12874:
12871:
12867:
12863:
12858:
12855:
12851:
12830:
12827:
12824:
12804:
12782:
12779:
12776:
12766:
12761:
12758:
12748:
12741:
12736:
12732:
12711:
12708:
12699:
12674:
12671:
12668:
12658:
12649:
12643:
12640:
12620:
12606:
12586:
12576:
12564:
12556:
12546:
12535:
12529:
12519:
12508:
12499:
12493:
12484:
12471:
12456:
12446:
12441:
12436:
12426:
12414:
12406:
12386:
12374:
12369:
12366:
12362:
12354:
12351:
12348:
12345:
12333:
12330:
12324:
12319:
12314:
12310:
12287:
12263:
12239:
12221:Special cases
12218:
12201:
12196:
12190:
12185:
12178:
12173:
12167:
12162:
12159:
12154:
12150:
12127:
12123:
12114:
12110:
12109:
12091:
12083:
12080:
12071:
12068:
12061:
12054:
12049:
12044:
12038:
12033:
12028:
12024:
12003:
11998:
11992:
11987:
11980:
11975:
11968:
11963:
11957:
11952:
11949:
11941:
11927:
11922:
11914:
11911:
11902:
11899:
11888:
11885:
11876:
11873:
11865:
11860:
11855:
11851:
11831:
11826:
11820:
11815:
11808:
11803:
11797:
11792:
11789:
11781:
11764:
11760:
11756:
11751:
11747:
11743:
11740:
11718:
11712:
11707:
11698:
11695:
11686:
11683:
11675:
11670:
11665:
11661:
11641:
11636:
11630:
11625:
11618:
11613:
11607:
11602:
11599:
11591:
11577:
11572:
11566:
11561:
11552:
11549:
11544:
11537:
11534:
11526:
11521:
11516:
11512:
11492:
11487:
11481:
11476:
11473:
11466:
11461:
11455:
11450:
11447:
11439:
11438:
11420:
11415:
11409:
11402:
11392:
11387:
11374:
11367:
11353:
11349:
11344:
11339:
11335:
11312:
11307:
11296:
11290:
11283:
11273:
11268:
11255:
11249:
11244:
11241:
11233:
11216:
11212:
11189:
11185:
11181:
11161:
11156:
11152:
11129:
11126:
11121:
11115:
11111:
11107:
11103:
11096:
11092:
11088:
11083:
11079:
11056:
11052:
11046:
11043:
11038:
11034:
11029:
11025:
11020:
11015:
11010:
11006:
10983:
10973:
10956:
10952:
10948:
10943:
10937:
10932:
10923:
10920:
10911:
10908:
10900:
10895:
10890:
10886:
10881:
10876:
10872:
10849:
10843:
10838:
10831:
10826:
10820:
10815:
10812:
10807:
10803:
10795:. Similarly,
10782:
10779:
10774:
10768:
10763:
10756:
10751:
10745:
10740:
10737:
10732:
10728:
10723:
10701:
10693:
10690:
10681:
10678:
10667:
10664:
10655:
10652:
10644:
10639:
10634:
10630:
10625:
10617:
10616:
10598:
10592:
10587:
10578:
10575:
10566:
10563:
10555:
10550:
10545:
10541:
10520:
10515:
10509:
10504:
10497:
10492:
10486:
10481:
10478:
10470:
10453:
10449:
10445:
10440:
10436:
10432:
10427:
10423:
10402:
10397:
10391:
10386:
10379:
10374:
10368:
10363:
10358:
10354:
10333:
10328:
10322:
10317:
10310:
10305:
10299:
10294:
10291:
10283:
10282:
10281:
10273:
10271:
10250:
10246:
10237:
10233:
10229:
10224:
10220:
10194:
10190:
10183:
10180:
10177:
10155:
10151:
10128:
10124:
10103:
10083:
10078:
10074:
10065:
10061:
10056:
10050:
10034:
10022:
10017:
10013:
10008:
10004:
10000:
9997:
9993:
9986:
9979:
9973:
9969:
9965:
9962:
9959:
9955:
9950:
9944:
9928:
9916:
9907:
9903:
9897:
9893:
9886:
9878:
9874:
9869:
9863:
9853:
9842:
9836:
9832:
9828:
9825:
9820:
9816:
9801:
9797:
9793:
9790:
9785:
9779:
9740:
9736:
9715:
9693:
9689:
9665:
9657:
9653:
9646:
9622:
9618:
9592:
9586:
9580:
9566:
9548:
9544:
9517:
9507:
9503:
9475:
9449:
9445:
9418:
9414:
9387:
9377:
9373:
9360:, say), then
9359:
9341:
9316:
9311:
9307:
9303:
9292:
9282:
9264:
9261:
9256:
9252:
9247:
9243:
9238:
9210:
9207:
9202:
9196:
9192:
9188:
9184:
9172:
9154:
9151:
9146:
9142:
9139:
9136:
9131:
9127:
9123:
9119:
9112:
9108:
9102:
9096:
9088:
9083:
9079:
9073:
9070:
9065:
9061:
9058:
9055:
9052:
9047:
9043:
9038:
9031:
9025:
9017:
9012:
9008:
8993:
8991:
8986:
8970:
8937:
8931:
8920:
8917:
8913:
8888:
8883:
8879:
8854:
8828:
8794:
8788:
8764:
8738:
8698:
8692:
8681:
8678:
8674:
8646:
8640:
8616:
8592:
8566:
8535:
8511:
8506:
8502:
8490:
8477:
8472:
8467:
8463:
8460:
8457:
8453:
8429:
8426:
8423:
8397:
8393:
8370:
8367:
8364:
8356:
8351:
8347:
8344:
8341:
8337:
8332:
8329:
8307:
8302:
8298:
8295:
8292:
8288:
8283:
8280:
8277:
8274:
8271:
8266:
8239:
8236:
8233:
8230:
8225:
8220:
8216:
8213:
8210:
8206:
8201:
8196:
8167:
8157:
8139:
8129:
8111:
8101:
8083:
8057:
8042:
8032:
8027:
8023:
7970:
7955:
7945:
7942:
7926:
7912:
7909:
7904:
7900:
7877:
7866:
7851:
7843:
7840:
7834:
7831:
7826:
7818:
7815:
7789:
7786:
7783:
7773:
7770:
7744:
7741:
7738:
7728:
7725:
7714:
7698:
7674:
7650:
7626:
7618:
7615:
7609:
7606:
7603:
7583:
7580:
7577:
7574:
7565:
7550:
7542:
7539:
7533:
7528:
7520:
7517:
7511:
7487:
7484:
7481:
7471:
7468:
7442:
7439:
7436:
7426:
7423:
7398:
7395:
7390:
7386:
7381:
7377:
7372:
7368:
7364:
7361:
7357:
7353:
7349:
7343:
7339:
7335:
7332:
7329:
7325:
7319:
7315:
7307:
7293:
7290:
7281:
7277:
7271:
7267:
7263:
7260:
7257:
7253:
7249:
7245:
7241:
7236:
7232:
7228:
7225:
7221:
7213:
7206:
7205:
7204:
7182:
7178:
7153:
7149:
7145:
7142:
7139:
7136:
7133:
7130:
7127:
7120:
7103:
7081:
7076:
7072:
7068:
7065:
7062:
7059:
7056:
7053:
7046:
7029:
7003:
6999:
6974:
6965:
6946:
6942:
6931:
6913:
6903:
6899:
6881:
6872:
6856:
6852:
6848:
6843:
6839:
6835:
6832:
6829:
6824:
6820:
6799:
6796:
6793:
6790:
6787:
6784:
6781:
6774:
6773:
6772:
6758:
6755:
6750:
6746:
6725:
6722:
6717:
6713:
6690:
6686:
6682:
6679:
6657:
6653:
6649:
6646:
6638:
6622:
6617:
6613:
6609:
6606:
6584:
6580:
6576:
6573:
6570:
6557:
6542:
6536:
6530:
6525:
6518:
6513:
6507:
6499:
6493:
6487:
6482:
6475:
6470:
6464:
6458:
6453:
6447:
6439:
6436:
6426:
6418:
6415:
6406:
6399:
6390:
6387:
6379:
6369:
6366:
6358:
6352:
6347:
6342:
6336:
6328:
6325:
6315:
6307:
6304:
6295:
6289:
6283:
6277:
6269:
6266:
6256:
6248:
6245:
6236:
6231:
6226:
6220:
6214:
6209:
6202:
6197:
6191:
6185:
6180:
6166:
6160:
6155:
6148:
6143:
6137:
6130:
6124:
6119:
6112:
6107:
6101:
6084:
6066:
6062:
6056:
6052:
6048:
6043:
6035:
6032:
6018:
6001:
5996:
5993:
5989:
5983:
5979:
5975:
5973:
5966:
5961:
5957:
5952:
5948:
5943:
5934:
5929:
5925:
5919:
5916:
5912:
5908:
5906:
5899:
5894:
5888:
5884:
5880:
5876:
5844:
5839:
5835:
5823:
5819:
5815:
5809:
5801:
5797:
5793:
5784:
5779:
5775:
5764:
5763:
5762:
5744:
5740:
5734:
5730:
5726:
5721:
5713:
5710:
5679:
5675:
5671:
5668:
5661:
5644:
5640:
5636:
5633:
5626:
5611:
5608:
5603:
5599:
5595:
5573:
5551:
5548:
5545:
5540:
5536:
5513:
5504:
5487:
5483:
5479:
5474:
5470:
5466:
5463:
5458:
5454:
5450:
5428:
5419:
5402:
5398:
5394:
5391:
5386:
5382:
5378:
5375:
5370:
5366:
5343:
5334:
5333:
5332:
5314:
5310:
5304:
5300:
5296:
5291:
5283:
5280:
5248:
5243:
5235:
5232:
5226:
5223:
5218:
5214:
5210:
5208:
5201:
5197:
5191:
5187:
5183:
5176:
5173:
5170:
5165:
5157:
5154:
5148:
5145:
5143:
5138:
5135:
5130:
5126:
5114:
5100:
5095:
5091:
5085:
5081:
5077:
5074:
5069:
5065:
5061:
5058:
5053:
5049:
5043:
5039:
5035:
5032:
5027:
5023:
5015:
4997:
4992:
4988:
4984:
4981:
4976:
4972:
4968:
4966:
4959:
4954:
4948:
4944:
4940:
4937:
4932:
4928:
4923:
4914:
4909:
4905:
4901:
4898:
4893:
4889:
4885:
4883:
4876:
4871:
4865:
4861:
4857:
4854:
4849:
4845:
4840:
4827:
4809:
4806:
4803:
4798:
4794:
4790:
4788:
4783:
4780:
4775:
4771:
4765:
4761:
4757:
4750:
4745:
4741:
4735:
4731:
4727:
4724:
4722:
4715:
4711:
4705:
4701:
4697:
4694:
4689:
4685:
4673:
4657:
4653:
4647:
4643:
4639:
4634:
4626:
4623:
4613:
4612:
4611:
4593:
4590:
4587:
4577:
4574:
4568:
4563:
4560:
4557:
4547:
4544:
4518:
4514:
4508:
4504:
4500:
4495:
4487:
4484:
4472:The equality
4465:
4447:
4443:
4439:
4415:
4410:
4406:
4394:
4381:
4376:
4371:
4365:
4361:
4357:
4353:
4346:
4342:
4338:
4333:
4329:
4309:
4304:
4300:
4294:
4289:
4285:
4280:
4276:
4271:
4266:
4261:
4257:
4242:
4229:
4224:
4220:
4214:
4207:
4203:
4197:
4190:
4186:
4180:
4173:
4169:
4163:
4158:
4154:
4133:
4111:
4107:
4086:
4081:
4077:
4069:
4066:
4062:
4054:
4050:
4044:
4037:
4033:
4025:
4022:
4018:
4010:
4006:
4000:
3995:
3991:
3970:
3948:
3944:
3923:
3920:
3913:
3909:
3901:
3898:
3894:
3888:
3881:
3878:
3874:
3866:
3862:
3856:
3851:
3848:
3818:
3813:
3809:
3805:
3801:
3798:
3795:
3791:
3786:
3782:
3778:
3774:
3771:
3750:
3745:
3741:
3737:
3733:
3730:
3727:
3723:
3718:
3714:
3710:
3706:
3703:
3695:
3678:
3675:
3672:
3648:
3644:
3638:
3635:
3631:
3627:
3622:
3617:
3613:
3610:
3606:
3580:
3576:
3551:
3541:
3527:
3522:
3517:
3512:
3508:
3504:
3499:
3494:
3489:
3484:
3480:
3476:
3470:
3460:
3456:
3450:
3445:
3440:
3432:
3427:
3421:
3411:
3406:
3402:
3398:
3393:
3388:
3383:
3374:
3370:
3360:
3345:
3342:
3337:
3332:
3327:
3323:
3319:
3309:
3292:
3289:
3285:
3281:
3276:
3272:
3263:
3245:
3235:
3216:
3212:
3187:
3177:
3176:
3175:
3173:
3163:
3161:
3159:
3139:
3135:
3131:
3126:
3122:
3118:
3113:
3109:
3088:
3085:
3082:
3077:
3073:
3069:
3060:
3042:
3038:
3013:
2992:
2978:
2956:
2952:
2948:
2945:
2942:
2937:
2933:
2910:
2906:
2902:
2899:
2894:
2890:
2886:
2866:
2844:
2840:
2817:
2813:
2807:
2803:
2799:
2796:
2791:
2787:
2766:
2746:
2743:
2740:
2718:
2714:
2710:
2707:
2704:
2701:
2693:
2673:
2670:
2665:
2661:
2657:
2649:
2648:right inverse
2633:
2628:
2625:
2620:
2614:
2610:
2606:
2602:
2595:
2591:
2587:
2582:
2578:
2553:
2549:
2522:
2518:
2514:
2492:
2472:
2464:
2449:
2446:
2443:
2438:
2434:
2425:
2409:
2404:
2400:
2394:
2391:
2386:
2382:
2377:
2373:
2368:
2363:
2358:
2354:
2329:
2325:
2300:
2295:
2291:
2266:
2242:
2232:
2231:
2230:
2212:
2208:
2180:
2177:
2174:
2158:
2142:
2120:
2117:
2114:
2104:
2101:
2079:
2075:
2065:
2049:
2045:
2041:
2036:
2032:
2025:
2020:
2016:
1990:
1986:
1965:
1960:
1956:
1935:
1932:
1929:
1921:
1917:
1913:
1890:
1868:
1864:
1860:
1838:
1834:
1830:
1810:
1805:
1801:
1780:
1760:
1738:
1734:
1730:
1710:
1688:
1684:
1663:
1658:
1654:
1633:
1628:
1624:
1620:
1615:
1607:
1602:
1598:
1572:
1568:
1564:
1561:
1556:
1546:
1542:
1538:
1513:
1509:
1505:
1485:
1480:
1476:
1452:
1449:
1444:
1440:
1435:
1430:
1425:
1421:
1416:
1412:
1407:
1383:
1378:
1374:
1364:
1350:
1345:
1341:
1337:
1333:
1328:
1323:
1317:
1313:
1309:
1305:
1295:
1275:
1271:
1267:
1258:
1244:
1239:
1235:
1230:
1225:
1221:
1217:
1212:
1208:
1199:
1179:
1175:
1165:
1151:
1148:
1144:
1141:
1136:
1132:
1128:
1100:
1096:
1092:
1083:
1082:
1081:
1063:
1060:
1057:
1047:
1042:
1038:
1009:
1006:
1003:
993:
990:
957:
954:
951:
941:
936:
932:
921:
904:
901:
898:
874:
864:
847:
837:
816:
810:
807:
783:
759:
749:
745:
741:
720:
714:
711:
685:
682:
679:
669:
666:
656:
635:
631:
626:
622:
596:
567:
563:
553:) is denoted
552:
528:
501:
498:
495:
485:
482:
472:
453:
450:
447:
390:
387:
384:
322:
293:
292:
291:
283:
281:
263:
239:
229:
225:
206:
202:
191:
190:normal matrix
173:
163:
159:
155:
150:
148:
144:
140:
136:
135:least squares
131:
129:
125:
124:
119:
118:pseudoinverse
115:
111:
107:
106:Roger Penrose
104:in 1951, and
103:
99:
95:
91:
90:pseudoinverse
73:
64:
44:
40:
30:
26:
22:
21030:Quantum mind
20894:
20855:Spin network
20827:
20809:
20795:
20775:
20767:
20759:
20751:
20743:
20558:Key concepts
20499:
20480:
20416:
20395:
20374:
20345:
20302:
20298:
20288:
20282:. CRC Press.
20279:
20245:
20241:
20199:
20195:
20156:(1): 17–19.
20153:
20147:
20141:
20127:
20113:
20096:
20092:
20086:
20069:
20065:
20059:
20041:
20024:
20018:
19988:(1): 61–74.
19985:
19981:
19975:
19939:
19935:
19929:
19918:
19893:
19887:
19883:
19874:
19847:
19841:
19800:
19794:
19775:
19769:
19736:
19732:
19726:
19717:
19713:
19700:
19665:
19661:
19655:
19622:
19618:
19608:
19584:
19561:
19536:
19485:
19479:
19454:
19450:
19441:
19422:
19416:
19409:Moore, E. H.
19403:
19389:, p. 42
19383:, p. 10
19367:
19310:
19143:
18997:automorphism
18987:
18885:between two
18834:
18802:
18726:
18547:
18354:
18191:
18119:
18116:
18086:
18083:
18071:denotes the
17829:
17627:denotes the
17350:
17263:
17249:Applications
17236:
17221:
17198:
17171:
17152:
17143:
16995:
16494:
16410:
16399:
16319:
16311:
16300:
16296:
16292:
16129:
16023:is given by
15826:
15690:
15629:
15362:
15356:followed by
15351:
15010:
14867:
14697:are of rank
14455:
14447:Construction
14251:
14118:
13943:
13883:
13723:
13617:
13400:
13294:
13077:
12612:
12415:
12412:
12229:
12216:
12113:left inverse
10279:
9572:
9288:
8999:
8987:
8491:
7932:
7868:Finally, if
7867:
7715:
7566:
7413:
7202:
6563:
6085:
6019:
5862:
5697:
5624:), or
5502:), or
5417:), or
5267:
4471:
4395:
4248:
3840:
3169:
3157:
3156:generalized
3061:
3003:
2689:
2647:
2424:left inverse
2423:
2066:
1467:
1198:weak inverse
1196:acts like a
982:
834:denotes the
744:column space
289:
151:
132:
121:
117:
89:
28:
18:
19619:SIAM Review
19377:, p. 7
18729:matrix norm
17182:linalg.pinv
16550:satisfying
16317:) is used.
15108:. Then the
14501:denote the
14021:such that
13940:EP matrices
12302:otherwise:
12254:is zero if
11327:, and thus
10864:, and thus
10716:, and thus
2426:, that is,
922:is denoted
224:annihilates
98:E. H. Moore
21:mathematics
21045:Categories
20600:algorithms
20466:PlanetMath
20451:PlanetMath
20331:References
19865:11093/6146
19334:Hat matrix
18994:involutive
17926:, we have
17480:, we have
17258:See also:
16284:GNU Octave
15931:such that
13506:rows, and
12142:, indeed,
10974:Note that
10170:(that is,
9569:Derivative
9291:continuous
9285:Continuity
8526:for given
8100:direct sum
6560:Projectors
3837:Identities
3262:invertible
2995:Properties
2646:This is a
1468:Note that
979:Definition
280:orthogonal
21009:(brother)
21003:(brother)
20834:) (1999)
20804:) (1996)
20627:CPU cache
20501:MathWorld
20482:MathWorld
20379:. Dover.
20321:0024-3795
20262:126385532
20224:125601192
20178:122260851
20033:841706164
19720:: 163–72.
19683:10217/536
19647:0036-1445
19267:
19215:
19108:∗
19076:∗
19060:
19041:∗
19028:
19010:
18861:→
18768:‖
18762:‖
18698:‖
18691:‖
18688:≤
18677:‖
18670:‖
18574:×
18561:∈
18526:‖
18519:‖
18516:≤
18507:‖
18500:‖
18405:‖
18398:‖
18292:−
18157:−
18051:‖
18047:⋅
18044:‖
17980:‖
17973:−
17964:‖
17961:≥
17950:‖
17943:−
17934:‖
17907:×
17894:∈
17888:∀
17856:×
17843:∈
17768:−
17669:−
17609:‖
17605:⋅
17602:‖
17532:−
17517:≥
17500:−
17454:∈
17448:∀
17298:×
17285:∈
17093:∗
17038:∗
16974:∗
16961:−
16949:δ
16938:∗
16892:∗
16728:σ
16685:σ
16670:α
16642:∗
16634:α
16599:∗
16455:−
16354:Σ
16286:function
16264:Σ
16236:∗
16222:Σ
16159:∗
16151:Σ
16103:∗
16084:∗
16071:∗
16055:∗
15979:∗
15947:∗
15885:∗
15853:×
15840:∈
15803:∗
15747:−
15736:∗
15717:∗
15669:∗
15608:∗
15587:∗
15577:∗
15550:∗
15523:∗
15463:∗
15378:∗
15336:∗
15310:∗
15301:⇔
15295:∗
15264:∗
15250:⇔
15244:∗
15231:−
15217:∗
15140:∗
15124:∗
15094:∗
15081:−
15067:∗
14961:∗
14930:∗
14881:∈
14846:∗
14833:−
14819:∗
14800:−
14789:∗
14770:∗
14678:×
14665:∈
14633:×
14620:∈
14531:×
14518:∈
14469:≤
14423:∗
14411:⋅
14402:Σ
14398:⋅
14362:∗
14350:⋅
14347:Σ
14344:⋅
14162:∗
13860:∗
13834:), then:
13807:∗
13761:∗
13673::
13596:−
13585:∗
13566:∗
13524:∗
13462:≤
13350::
13276:∗
13263:−
13249:∗
13198:∗
13139:≤
13015:⟹
13008:≠
12972:×
12917:∈
12868:δ
12828:×
12780:×
12767:⊕
12759:−
12752:~
12703:~
12672:×
12659:⊕
12653:~
12583:otherwise
12568:→
12557:∗
12550:→
12530:∗
12523:→
12503:→
12488:→
12460:→
12430:→
12382:otherwise
12367:−
11545:−
11474:−
11421:∗
11313:∗
11190:∗
11157:∗
11127:−
11116:∗
11097:∗
11057:∗
11044:−
11030:∗
10069:⊤
10039:⊤
10001:−
9963:−
9933:⊤
9911:⊤
9829:−
9650:↦
9584:↦
9262:−
9248:∗
9208:−
9197:∗
9152:−
9140:δ
9132:∗
9113:∗
9100:↘
9097:δ
9084:∗
9071:−
9059:δ
9048:∗
9029:↘
9026:δ
8918:−
8779:sends to
8679:−
8473:⊥
8461:
8427:
8368:
8362:→
8357:⊥
8345:
8308:⊥
8296:
8284:⊕
8278:
8237:
8231:⊕
8226:⊥
8214:
8112:⊥
8084:⊕
8048:→
7961:→
7787:×
7774:∈
7742:×
7729:∈
7485:×
7472:∈
7440:×
7427:∈
7391:∗
7365:−
7333:−
7320:∗
7261:−
7229:−
7183:∗
7143:−
7131:−
7069:−
7057:−
7004:∗
6947:∗
6900:onto the
6691:∗
6658:∗
6290:≠
6049:≠
5997:∗
5953:∗
5920:∗
5889:∗
5645:∗
5459:∗
5371:∗
5219:∗
5202:∗
5096:∗
5086:∗
5054:∗
5044:∗
4977:∗
4960:∗
4933:∗
4910:∗
4877:∗
4866:∗
4799:∗
4776:∗
4746:∗
4736:∗
4716:∗
4706:∗
4591:×
4578:∈
4561:×
4548:∈
4448:∗
4411:∗
4366:∗
4347:∗
4305:∗
4281:∗
4225:∗
4174:∗
4159:∗
4112:∗
4070:∗
4055:∗
4038:∗
4026:∗
3914:∗
3902:∗
3882:∗
3867:∗
3814:∗
3802:
3775:
3746:∗
3734:
3707:
3676:≠
3673:α
3636:−
3632:α
3611:α
3523:∗
3485:∗
3466:¯
3436:¯
3290:−
3158:reflexive
2957:∗
2911:∗
2818:∗
2719:∗
2626:−
2615:∗
2596:∗
2523:∗
2405:∗
2392:−
2378:∗
2296:∗
2118:×
2105:∈
1431:∗
1329:∗
1294:Hermitian
1061:×
1048:∈
1007:×
994:∈
955:×
942:∈
902:×
811:
715:
683:×
670:∈
627:∗
568:∗
499:×
486:∈
451:×
388:×
147:Euclidean
100:in 1920,
21015:(sister)
20997:(father)
20843:Concepts
20656:Software
20620:Hardware
20579:Problems
20344:(2003).
19692:17660144
19533:(1996).
19474:(1955).
19411:(1920).
19323:See also
19095:, where
18785:‖
18772:‖
17539:‖
17522:‖
17507:‖
17490:‖
17234:method.
17230:and the
17178:matrix.I
16303:)⋅max(Σ)
16295:= ε⋅max(
16254:such as
16250:. For a
15430:, where
15157:, where
12478:if
12341:if
10618:Indeed,
10276:Examples
9356:(in the
8154:for the
8126:for the
8098:for the
2832:, where
2155:is full
2005:implies
1903:implies
286:Notation
20988:Related
20216:3617890
20158:Bibcode
20010:2156431
19990:Bibcode
19964:2949637
19944:Bibcode
19910:2038377
19761:2156365
19741:Bibcode
19627:Bibcode
19490:Bibcode
19144:Example
18983:
18956:
18948:
18921:
18917:
18890:
18883:
18837:
18825:
18805:
18591:
18550:
18341:
18321:
18317:
18281:
18277:
18257:
18214:
18194:
17924:
17880:
17873:
17832:
17825:
17802:
17798:
17752:
17748:
17728:
17724:
17704:
17478:
17440:
17373:
17353:
17315:
17274:
17134:
17114:
17110:
17080:
17076:
17056:
17052:
17022:
17018:
16998:
16841:
16814:
16810:
16783:
16779:
16759:
16755:
16719:
16717:, with
16656:(where
16548:
16521:
16517:
16497:
16390:
16370:
16366:
16346:
16342:
16322:
16276:
16256:
16197:, then
16195:
16175:
16021:
16001:
15870:
15829:
15817:
15790:
15786:
15766:
15686:
15656:
15652:
15632:
15508:
15488:
15395:
15365:
15179:
15159:
15033:
15013:
15006:
14986:
14978:
14948:
14944:
14914:
14721:. Then
14719:
14699:
14695:
14654:
14650:
14609:
14574:can be
14572:
14552:
14550:. Then
14548:
14507:
14499:
14458:
14315:; then
14311:be the
14309:
14285:
14277:
14257:
14141:
14121:
14109:
14089:
14085:
14065:
13990:
13970:
13966:
13946:
13934:
13914:
13906:
13886:
13746:
13726:
13671:
13651:
13647:
13620:
13538:
13508:
13501:
13481:
13477:
13451:
13449:, (for
13447:
13427:
13423:
13403:
13348:
13328:
13324:
13297:
13215:
13185:
13178:
13158:
13154:
13128:
13126:, (for
13124:
13104:
13100:
13080:
12934:, then
12817:is any
12724:, then
12409:Vectors
12300:
12280:
12276:
12256:
12252:
12232:
12226:Scalars
10996:
10976:
9637:
9610:
9563:
9536:
9532:
9492:
9488:
9468:
9464:
9437:
9433:
9406:
9402:
9362:
9354:
9334:
9330:
9295:
9279:
9229:
9225:
9175:
8983:
8963:
8959:
8905:
8901:
8871:
8867:
8847:
8843:
8814:
8810:
8781:
8777:
8757:
8753:
8724:
8720:
8666:
8662:
8633:
8629:
8609:
8605:
8585:
8581:
8552:
8548:
8528:
8524:
8494:
8442:
8416:
8412:
8385:
8180:
8160:
8152:
8132:
8124:
8104:
8096:
8076:
8072:
8015:
8011:
7989:
7985:
7935:
7890:
7870:
7804:
7763:
7759:
7718:
7711:
7691:
7687:
7667:
7663:
7643:
7502:
7461:
7457:
7416:
7197:
7170:
7116:
7096:
7042:
7022:
7018:
6991:
6987:
6967:
6961:
6934:
6926:
6906:
6896:is the
6894:
6874:
6081:
6022:
5759:
5700:
5586:
5566:
5526:
5506:
5441:
5421:
5356:
5336:
5329:
5270:
4608:
4537:
4533:
4474:
4462:
4432:
4428:
4398:
3691:
3665:
3595:
3568:
3564:
3544:
3258:
3238:
3231:
3204:
3200:
3180:
3160:inverse
3057:
3030:
3026:
3006:
2568:
2541:
2537:
2507:
2344:
2317:
2313:
2283:
2279:
2259:
2255:
2235:
2227:
2200:
2198:, then
2196:
2161:
1723:), and
1396:
1366:
1290:
1260:
1194:
1167:
1115:
1085:
1078:
1030:
972:
924:
917:
891:
887:
867:
860:
840:
832:
800:
796:
776:
772:
752:
736:
704:
700:
659:
614:, then
582:
555:
547:
520:
516:
475:
468:
433:
429:
407:
403:
377:
373:
351:
347:
325:
317:
295:
276:
256:
252:
232:
221:
194:
186:
166:
158:complex
86:
66:
59:
32:
20830:(with
20814:(with
20800:(with
20780:(2016)
20772:(2010)
20764:(2004)
20756:(1994)
20748:(1989)
20678:LAPACK
20668:MATLAB
20429:
20402:
20383:
20360:
20319:
20260:
20222:
20214:
20176:
20031:
20008:
19962:
19908:
19782:
19759:
19690:
19645:
19595:
19549:
19545:–258.
19256:while
18952:closed
18000:where
17558:where
17272:. For
17243:pinv()
17184:; its
17155:LAPACK
16315:LAPACK
16280:MATLAB
15486:, and
15181:is an
14607:where
14252:For a
13910:normal
9990:
9984:
9890:
9884:
8156:kernel
7288:
7285:
7218:
5564:) and
4572:
4146:gives
3983:gives
1948:, and
889:, the
836:kernel
798:) and
321:fields
228:kernel
63:matrix
27:, the
20737:Books
20663:ATLAS
20258:S2CID
20220:S2CID
20212:JSTOR
20174:S2CID
20051:(PDF)
20006:JSTOR
19960:JSTOR
19906:JSTOR
19757:JSTOR
19710:(PDF)
19688:S2CID
19360:Notes
18990:field
17190:SciPy
17174:NumPy
16134:. If
12961:is a
11144:, as
9169:(see
8755:that
8013:then
6902:range
2650:, as
2233:When
750:) of
748:image
740:range
584:. If
188:is a
61:of a
20642:SIMD
20427:ISBN
20400:ISBN
20381:ISBN
20358:ISBN
20317:ISSN
20029:OCLC
19780:ISBN
19643:ISSN
19593:ISBN
19547:ISBN
19264:rank
19212:rank
19057:rank
19025:rank
19007:rank
18919:and
18745:cond
18036:and
17594:and
17228:pinv
17222:The
17217:pinv
17209:ginv
17205:ginv
17186:pinv
17180:and
17054:for
16673:<
16667:<
16288:pinv
15788:and
14868:For
14652:and
14503:rank
14456:Let
14176:and
12709:>
12689:and
11942:For
11782:For
11592:For
11440:For
11174:and
11071:nor
10471:For
10284:For
9573:Let
8252:and
6812:and
6738:and
6635:are
6599:and
5659:, or
4430:and
4126:for
3963:for
3764:and
3663:for
2925:and
2157:rank
1823:and
1587:and
1498:and
983:For
657:For
473:For
226:the
154:real
120:and
20632:TLB
20464:at
20449:at
20423:240
20350:doi
20307:doi
20250:doi
20204:doi
20166:doi
20101:doi
20074:doi
19998:doi
19969:pdf
19952:doi
19898:doi
19860:hdl
19852:doi
19848:128
19805:doi
19749:doi
19678:hdl
19670:doi
19635:doi
19543:257
19498:doi
19427:doi
19311:In
18954:in
18597:If
18427:If
17237:In
17213:svd
17165:or
16282:or
15630:so
15401:of
14984:of
14946:or
14578:as
14505:of
14472:min
13908:is
13884:If
9755::
9728:at
9681:at
9227:or
9093:lim
9022:lim
8550:in
8458:ran
8424:ran
8414:on
8365:ran
8342:ker
8293:ran
8275:ran
8234:ker
8211:ker
8168:ran
8140:ker
6904:of
4396:as
3799:ran
3772:ran
3731:ker
3704:ker
3260:is
3236:If
3178:If
2169:min
1292:is
808:ker
712:ran
230:of
156:or
19:In
21047::
20818:,
20498:.
20479:.
20425:.
20356:.
20340:;
20315:.
20301:.
20297:.
20270:^
20256:.
20246:62
20244:.
20232:^
20218:.
20210:.
20200:63
20198:.
20186:^
20172:.
20164:.
20154:52
20152:.
20097:24
20095:.
20070:25
20068:.
20004:.
19996:.
19986:11
19984:.
19958:.
19950:.
19938:.
19904:.
19894:34
19892:.
19858:.
19846:.
19832:^
19817:^
19803:.
19755:.
19747:.
19737:10
19735:.
19718:49
19716:.
19712:.
19686:.
19676:.
19664:.
19641:.
19633:.
19621:.
19617:.
19591:.
19573:^
19529:;
19512:^
19496:.
19486:51
19484:.
19478:.
19463:^
19455:49
19453:.
19423:26
19421:.
19415:.
18985:.
18593:.
17875:.
16924::=
16309:.
16299:,
15819:.
15688:.
15360:.
14860:.
14111:.
13716:.
13393:.
13070:.
12899:,
10230::=
10212:,
10181::=
10116:,
9565:.
9490:,
8992:.
8985:.
8130:,
8102:,
7925:.
7596:,
7044:).
6963:).
6672:,
6083::
5761::
5331::
3174:.
3059:.
2064:.
1296::
1200::
1119:A
702:,
349:,
130:.
20957:/
20722:e
20715:t
20708:v
20602:)
20598:(
20543:e
20536:t
20529:v
20504:.
20485:.
20468:.
20453:.
20435:.
20408:.
20389:.
20366:.
20352::
20323:.
20309::
20303:1
20264:.
20252::
20226:.
20206::
20180:.
20168::
20160::
20135:.
20121:.
20107:.
20103::
20080:.
20076::
20053:.
20035:.
20012:.
20000::
19992::
19966:.
19954::
19946::
19940:3
19912:.
19900::
19884:r
19868:.
19862::
19854::
19827:.
19811:.
19807::
19788:.
19763:.
19751::
19743::
19694:.
19680::
19672::
19666:4
19649:.
19637::
19629::
19623:8
19603:.
19601:.
19568:.
19555:.
19506:.
19500::
19492::
19457:.
19435:.
19429::
19296:0
19293:=
19289:)
19285:A
19280:T
19276:A
19271:(
19244:1
19241:=
19237:)
19231:T
19227:A
19223:A
19219:(
19190:T
19184:]
19178:i
19173:1
19167:[
19161:=
19158:A
19130:A
19104:A
19082:)
19072:A
19068:A
19064:(
19054:=
19050:)
19046:A
19037:A
19032:(
19022:=
19019:)
19016:A
19013:(
18969:2
18965:H
18934:2
18930:H
18903:1
18899:H
18869:2
18865:H
18856:1
18852:H
18848::
18845:A
18813:A
18789:.
18780:+
18776:A
18765:A
18759:=
18756:)
18753:A
18750:(
18703:F
18694:X
18682:F
18673:Z
18650:B
18645:+
18641:A
18637:=
18634:Z
18614:B
18611:=
18608:X
18605:A
18577:p
18571:m
18566:K
18558:B
18530:2
18522:x
18511:2
18503:z
18480:b
18475:+
18471:A
18467:=
18464:z
18444:b
18441:=
18438:x
18435:A
18409:2
18401:x
18375:,
18372:b
18369:=
18366:x
18363:A
18329:A
18305:A
18300:+
18296:A
18289:I
18265:A
18243:b
18240:=
18237:b
18232:+
18228:A
18224:A
18202:w
18178:w
18174:]
18170:A
18165:+
18161:A
18154:I
18150:[
18146:+
18143:b
18138:+
18134:A
18130:=
18127:x
18103:b
18100:=
18097:x
18094:A
18075:.
18056:F
18024:B
18019:+
18015:A
18011:=
18008:Z
17985:F
17976:B
17970:Z
17967:A
17955:F
17946:B
17940:X
17937:A
17910:p
17904:n
17899:K
17891:X
17859:p
17853:m
17848:K
17840:B
17813:.
17810:z
17785:)
17781:A
17776:+
17772:A
17765:I
17761:(
17736:A
17712:w
17690:w
17686:)
17682:A
17677:+
17673:A
17666:I
17662:(
17658:+
17655:b
17650:+
17646:A
17642:=
17639:x
17613:2
17582:b
17577:+
17573:A
17569:=
17566:z
17544:2
17535:b
17529:z
17526:A
17512:2
17503:b
17497:x
17494:A
17464:n
17459:K
17451:x
17423:A
17403:A
17383:b
17361:x
17337:,
17334:b
17331:=
17328:x
17325:A
17301:n
17295:m
17290:K
17282:A
17201:R
17122:A
17098:A
17089:A
17064:A
17034:A
17030:A
17006:A
16970:A
16964:1
16956:)
16952:I
16946:+
16943:A
16934:A
16929:(
16919:0
16915:A
16887:)
16883:A
16878:0
16874:A
16869:(
16864:=
16861:A
16856:0
16852:A
16827:0
16823:A
16796:i
16792:A
16767:A
16743:)
16740:A
16737:(
16732:1
16705:)
16702:A
16699:(
16694:2
16689:1
16680:/
16676:2
16664:0
16638:A
16631:=
16626:0
16622:A
16594:)
16590:A
16585:0
16581:A
16576:(
16571:=
16568:A
16563:0
16559:A
16534:0
16530:A
16505:A
16481:,
16476:i
16472:A
16468:A
16463:i
16459:A
16450:i
16446:A
16442:2
16439:=
16434:1
16431:+
16428:i
16424:A
16378:A
16330:A
16301:n
16297:m
16293:t
16232:U
16226:+
16218:V
16215:=
16210:+
16206:A
16183:A
16155:V
16148:U
16145:=
16142:A
16111:.
16108:)
16099:A
16095:A
16092:(
16089:p
16080:A
16076:=
16067:A
16063:)
16060:A
16051:A
16047:(
16044:p
16041:=
16036:+
16032:A
16009:A
15987:)
15984:A
15975:A
15971:(
15968:p
15965:=
15960:+
15956:)
15952:A
15943:A
15939:(
15919:)
15916:t
15913:(
15910:p
15890:A
15881:A
15856:n
15850:m
15845:K
15837:A
15799:A
15774:A
15750:1
15742:)
15732:A
15728:A
15724:(
15713:A
15709:=
15704:+
15700:A
15674:A
15665:A
15640:R
15616:,
15613:R
15604:R
15599:=
15595:R
15592:Q
15583:Q
15573:R
15568:=
15564:)
15561:R
15558:Q
15555:(
15546:)
15542:R
15539:Q
15536:(
15532:=
15528:A
15519:A
15496:R
15474:I
15471:=
15468:Q
15459:Q
15438:Q
15418:R
15415:Q
15412:=
15409:A
15383:A
15374:A
15332:A
15328:=
15323:+
15319:A
15315:R
15306:R
15291:A
15287:=
15282:+
15278:A
15273:)
15269:A
15260:A
15255:(
15240:A
15234:1
15226:)
15222:A
15213:A
15208:(
15203:=
15198:+
15194:A
15167:R
15145:R
15136:R
15132:=
15129:A
15120:A
15090:A
15084:1
15076:)
15072:A
15063:A
15058:(
15053:=
15048:+
15044:A
15021:A
14994:A
14966:A
14957:A
14926:A
14922:A
14900:}
14896:C
14892:,
14888:R
14884:{
14877:K
14842:B
14836:1
14828:)
14824:B
14815:B
14810:(
14803:1
14795:)
14785:C
14781:C
14777:(
14766:C
14762:=
14757:+
14753:B
14747:+
14743:C
14739:=
14734:+
14730:A
14707:r
14681:n
14675:r
14670:K
14662:C
14636:r
14630:m
14625:K
14617:B
14595:C
14592:B
14589:=
14586:A
14560:A
14534:n
14528:m
14523:K
14515:A
14487:)
14484:n
14481:,
14478:m
14475:(
14466:r
14428:.
14417:F
14406:+
14393:F
14388:=
14379:+
14375:C
14367:,
14356:F
14339:F
14334:=
14327:C
14295:F
14265:C
14233:.
14230:A
14227:=
14222:+
14218:A
14197:A
14194:=
14189:2
14185:A
14158:A
14154:=
14151:A
14129:A
14097:A
14073:A
14051:)
14048:A
14045:(
14042:p
14039:=
14034:+
14030:A
14009:)
14006:t
14003:(
14000:p
13978:A
13954:A
13922:A
13894:A
13865:.
13856:A
13852:=
13847:+
13843:A
13820:m
13816:I
13812:=
13803:A
13799:A
13777:n
13773:I
13769:=
13766:A
13757:A
13734:A
13702:m
13698:I
13694:=
13689:+
13685:A
13681:A
13659:A
13633:+
13629:A
13604:.
13599:1
13591:)
13581:A
13577:A
13573:(
13562:A
13558:=
13553:+
13549:A
13520:A
13516:A
13489:m
13465:n
13459:m
13435:m
13411:A
13379:n
13375:I
13371:=
13368:A
13363:+
13359:A
13336:A
13310:+
13306:A
13281:.
13272:A
13266:1
13258:)
13254:A
13245:A
13240:(
13235:=
13230:+
13226:A
13203:A
13194:A
13166:n
13142:m
13136:n
13112:n
13088:A
13056:i
13053:i
13049:A
13044:/
13040:1
13037:=
13032:+
13027:i
13024:i
13020:A
13011:0
13003:i
13000:i
12996:A
12975:m
12969:n
12947:+
12943:A
12921:K
12912:i
12908:a
12885:i
12881:a
12875:j
12872:i
12864:=
12859:j
12856:i
12852:A
12831:n
12825:m
12805:A
12783:k
12777:k
12772:0
12762:1
12749:D
12742:=
12737:+
12733:D
12712:0
12700:D
12675:k
12669:k
12664:0
12650:D
12644:=
12641:D
12621:D
12587:.
12577:,
12574:)
12565:x
12547:x
12540:(
12536:/
12520:x
12509:;
12500:0
12494:=
12485:x
12472:,
12467:T
12457:0
12447:{
12442:=
12437:+
12427:x
12387:.
12375:,
12370:1
12363:x
12355:;
12352:0
12349:=
12346:x
12334:,
12331:0
12325:{
12320:=
12315:+
12311:x
12288:x
12264:x
12240:x
12202:.
12197:)
12191:1
12186:0
12179:0
12174:1
12168:(
12163:=
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12155:+
12151:A
12128:+
12124:A
12106:.
12092:)
12084:2
12081:1
12072:2
12069:1
12062:0
12055:0
12050:0
12045:1
12039:(
12034:=
12029:+
12025:A
12004:,
11999:)
11993:1
11988:0
11981:1
11976:0
11969:0
11964:1
11958:(
11953:=
11950:A
11928:.
11923:)
11915:4
11912:1
11903:4
11900:1
11889:4
11886:1
11877:4
11874:1
11866:(
11861:=
11856:+
11852:A
11832:,
11827:)
11821:1
11816:1
11809:1
11804:1
11798:(
11793:=
11790:A
11779:.
11765:2
11761:2
11757:+
11752:2
11748:1
11744:=
11741:5
11719:)
11713:0
11708:0
11699:5
11696:2
11687:5
11684:1
11676:(
11671:=
11666:+
11662:A
11642:,
11637:)
11631:0
11626:2
11619:0
11614:1
11608:(
11603:=
11600:A
11578:.
11573:)
11567:0
11562:0
11553:2
11550:1
11538:2
11535:1
11527:(
11522:=
11517:+
11513:A
11493:,
11488:)
11482:0
11477:1
11467:0
11462:1
11456:(
11451:=
11448:A
11435:.
11416:)
11410:2
11403:2
11398:e
11393:+
11388:1
11383:e
11375:(
11368:1
11363:e
11354:2
11350:1
11345:=
11340:+
11336:A
11308:1
11303:e
11297:)
11291:2
11284:2
11279:e
11274:+
11269:1
11264:e
11256:(
11250:2
11245:=
11242:A
11217:+
11213:A
11186:A
11182:A
11162:A
11153:A
11130:1
11122:)
11112:A
11108:A
11104:(
11093:A
11089:=
11084:+
11080:A
11053:A
11047:1
11039:)
11035:A
11026:A
11021:(
11016:=
11011:+
11007:A
10984:A
10971:.
10957:+
10953:A
10949:=
10944:)
10938:0
10933:0
10924:2
10921:1
10912:2
10909:1
10901:(
10896:=
10891:+
10887:A
10882:A
10877:+
10873:A
10850:)
10844:0
10839:0
10832:0
10827:1
10821:(
10816:=
10813:A
10808:+
10804:A
10783:A
10780:=
10775:)
10769:0
10764:1
10757:0
10752:1
10746:(
10741:=
10738:A
10733:+
10729:A
10724:A
10702:)
10694:2
10691:1
10682:2
10679:1
10668:2
10665:1
10656:2
10653:1
10645:(
10640:=
10635:+
10631:A
10626:A
10613:.
10599:)
10593:0
10588:0
10579:2
10576:1
10567:2
10564:1
10556:(
10551:=
10546:+
10542:A
10521:,
10516:)
10510:0
10505:1
10498:0
10493:1
10487:(
10482:=
10479:A
10454:+
10450:A
10446:A
10441:+
10437:A
10433:=
10428:+
10424:A
10403:.
10398:)
10392:0
10387:0
10380:0
10375:0
10369:(
10364:=
10359:+
10355:A
10334:,
10329:)
10323:0
10318:0
10311:0
10306:0
10300:(
10295:=
10292:A
10256:)
10251:0
10247:x
10243:(
10238:+
10234:A
10225:+
10221:A
10200:)
10195:0
10191:x
10187:(
10184:A
10178:A
10156:0
10152:x
10129:+
10125:A
10104:A
10084:,
10079:+
10075:A
10066:+
10062:A
10057:)
10051:x
10047:d
10035:A
10030:d
10023:(
10018:)
10014:A
10009:+
10005:A
9998:I
9994:(
9987:+
9980:)
9974:+
9970:A
9966:A
9960:I
9956:(
9951:)
9945:x
9941:d
9929:A
9924:d
9917:(
9908:+
9904:A
9898:+
9894:A
9887:+
9879:+
9875:A
9870:)
9864:x
9860:d
9854:A
9850:d
9843:(
9837:+
9833:A
9826:=
9821:+
9817:A
9802:0
9798:x
9794:=
9791:x
9786:|
9780:x
9776:d
9770:d
9741:0
9737:x
9716:A
9694:0
9690:x
9669:)
9666:x
9663:(
9658:+
9654:A
9647:x
9623:0
9619:x
9596:)
9593:x
9590:(
9587:A
9581:x
9549:+
9545:A
9518:+
9514:)
9508:n
9504:A
9500:(
9476:A
9450:n
9446:A
9419:+
9415:A
9388:+
9384:)
9378:n
9374:A
9370:(
9342:A
9317:)
9312:n
9308:A
9304:(
9265:1
9257:)
9253:A
9244:A
9239:(
9211:1
9203:)
9193:A
9189:A
9185:(
9155:1
9147:)
9143:I
9137:+
9128:A
9124:A
9120:(
9109:A
9103:0
9089:=
9080:A
9074:1
9066:)
9062:I
9056:+
9053:A
9044:A
9039:(
9032:0
9018:=
9013:+
9009:A
8971:A
8947:)
8944:}
8941:)
8938:b
8935:(
8932:p
8929:{
8926:(
8921:1
8914:A
8889:b
8884:+
8880:A
8855:A
8829:n
8824:K
8798:)
8795:b
8792:(
8789:p
8765:A
8739:n
8734:K
8708:)
8705:}
8702:)
8699:b
8696:(
8693:p
8690:{
8687:(
8682:1
8675:A
8650:)
8647:b
8644:(
8641:p
8617:A
8593:b
8567:m
8562:K
8536:b
8512:b
8507:+
8503:A
8478:.
8468:)
8464:A
8454:(
8430:A
8398:+
8394:A
8371:A
8352:)
8348:A
8338:(
8333::
8330:A
8303:)
8299:A
8289:(
8281:A
8272:=
8267:m
8262:K
8240:A
8221:)
8217:A
8207:(
8202:=
8197:n
8192:K
8058:n
8053:K
8043:m
8038:K
8033::
8028:+
8024:A
7998:K
7971:m
7966:K
7956:n
7951:K
7946::
7943:A
7913:A
7910:=
7905:+
7901:A
7878:A
7852:+
7848:)
7844:B
7841:A
7838:(
7835:=
7832:A
7827:+
7823:)
7819:B
7816:A
7813:(
7790:m
7784:n
7779:K
7771:B
7745:n
7739:n
7734:K
7726:A
7699:A
7675:C
7651:D
7627:+
7623:)
7619:A
7616:B
7613:(
7610:A
7607:=
7604:D
7584:A
7581:B
7578:=
7575:C
7551:+
7547:)
7543:A
7540:B
7537:(
7534:=
7529:+
7525:)
7521:A
7518:B
7515:(
7512:A
7488:n
7482:m
7477:K
7469:B
7443:n
7437:n
7432:K
7424:A
7399:0
7396:=
7387:A
7382:)
7378:A
7373:+
7369:A
7362:I
7358:(
7354:=
7350:)
7344:+
7340:A
7336:A
7330:I
7326:(
7316:A
7294:0
7291:=
7282:A
7278:)
7272:+
7268:A
7264:A
7258:I
7254:(
7250:=
7246:)
7242:A
7237:+
7233:A
7226:I
7222:(
7214:A
7199:.
7179:A
7154:+
7150:A
7146:A
7140:I
7137:=
7134:P
7128:I
7118:.
7104:A
7082:A
7077:+
7073:A
7066:I
7063:=
7060:Q
7054:I
7030:A
7000:A
6975:Q
6943:A
6914:A
6882:P
6857:+
6853:A
6849:=
6844:+
6840:A
6836:Q
6833:=
6830:P
6825:+
6821:A
6800:A
6797:=
6794:Q
6791:A
6788:=
6785:A
6782:P
6759:Q
6756:=
6751:2
6747:Q
6726:P
6723:=
6718:2
6714:P
6687:Q
6683:=
6680:Q
6654:P
6650:=
6647:P
6623:A
6618:+
6614:A
6610:=
6607:Q
6585:+
6581:A
6577:A
6574:=
6571:P
6543:+
6537:)
6531:0
6526:0
6519:1
6514:1
6508:(
6500:+
6494:)
6488:1
6483:1
6476:0
6471:0
6465:(
6459:=
6454:)
6448:0
6440:2
6437:1
6427:0
6419:2
6416:1
6407:(
6400:)
6391:2
6388:1
6380:0
6370:2
6367:1
6359:0
6353:(
6348:=
6343:)
6337:0
6329:4
6326:1
6316:0
6308:4
6305:1
6296:(
6284:)
6278:0
6270:2
6267:1
6257:0
6249:2
6246:1
6237:(
6232:=
6227:+
6221:)
6215:0
6210:0
6203:1
6198:1
6192:(
6186:=
6181:+
6175:)
6167:)
6161:1
6156:1
6149:0
6144:0
6138:(
6131:)
6125:0
6120:0
6113:1
6108:1
6102:(
6095:(
6067:+
6063:A
6057:+
6053:B
6044:+
6040:)
6036:B
6033:A
6030:(
6002:.
5994:+
5990:A
5984:+
5980:A
5976:=
5967:+
5962:)
5958:A
5949:A
5944:(
5935:,
5930:+
5926:A
5917:+
5913:A
5909:=
5900:+
5895:)
5885:A
5881:A
5877:(
5848:)
5845:A
5840:+
5836:A
5832:(
5829:)
5824:+
5820:B
5816:B
5813:(
5810:=
5807:)
5802:+
5798:B
5794:B
5791:(
5788:)
5785:A
5780:+
5776:A
5772:(
5745:+
5741:A
5735:+
5731:B
5727:=
5722:+
5718:)
5714:B
5711:A
5708:(
5694:.
5680:+
5676:A
5672:=
5669:B
5641:A
5637:=
5634:B
5612:I
5609:=
5604:+
5600:B
5596:B
5574:B
5552:I
5549:=
5546:A
5541:+
5537:A
5514:A
5488:n
5484:I
5480:=
5475:+
5471:B
5467:B
5464:=
5455:B
5451:B
5429:B
5403:n
5399:I
5395:=
5392:A
5387:+
5383:A
5379:=
5376:A
5367:A
5344:A
5315:+
5311:A
5305:+
5301:B
5297:=
5292:+
5288:)
5284:B
5281:A
5278:(
5249:.
5244:+
5240:)
5236:B
5233:A
5230:(
5227:B
5224:A
5215:A
5211:=
5198:A
5192:+
5188:B
5184:B
5177:,
5174:B
5171:A
5166:+
5162:)
5158:B
5155:A
5152:(
5149:B
5146:=
5139:B
5136:A
5131:+
5127:A
5101:A
5092:A
5082:B
5078:B
5075:=
5070:+
5066:B
5062:B
5059:A
5050:A
5040:B
5036:B
5033:A
5028:+
5024:A
4998:.
4993:+
4989:B
4985:B
4982:A
4973:A
4969:=
4955:)
4949:+
4945:B
4941:B
4938:A
4929:A
4924:(
4915:,
4906:B
4902:B
4899:A
4894:+
4890:A
4886:=
4872:)
4862:B
4858:B
4855:A
4850:+
4846:A
4841:(
4810:.
4807:B
4804:A
4795:A
4791:=
4784:B
4781:A
4772:A
4766:+
4762:B
4758:B
4751:,
4742:A
4732:B
4728:B
4725:=
4712:A
4702:B
4698:B
4695:A
4690:+
4686:A
4658:+
4654:A
4648:+
4644:B
4640:=
4635:+
4631:)
4627:B
4624:A
4621:(
4594:p
4588:n
4583:K
4575:B
4569:,
4564:n
4558:m
4553:K
4545:A
4519:+
4515:A
4509:+
4505:B
4501:=
4496:+
4492:)
4488:B
4485:A
4482:(
4444:A
4440:A
4416:A
4407:A
4382:,
4377:+
4372:)
4362:A
4358:A
4354:(
4343:A
4339:=
4334:+
4330:A
4310:,
4301:A
4295:+
4290:)
4286:A
4277:A
4272:(
4267:=
4262:+
4258:A
4230:.
4221:A
4215:A
4208:+
4204:A
4198:=
4191:+
4187:A
4181:A
4170:A
4164:=
4155:A
4134:A
4108:A
4087:,
4082:+
4078:A
4067:+
4063:A
4051:A
4045:=
4034:A
4023:+
4019:A
4011:+
4007:A
4001:=
3996:+
3992:A
3971:A
3949:+
3945:A
3924:.
3921:A
3910:A
3899:+
3895:A
3889:=
3879:+
3875:A
3863:A
3857:A
3852:=
3849:A
3832:.
3819:)
3810:A
3806:(
3796:=
3792:)
3787:+
3783:A
3779:(
3751:)
3742:A
3738:(
3728:=
3724:)
3719:+
3715:A
3711:(
3693:.
3679:0
3649:+
3645:A
3639:1
3628:=
3623:+
3618:)
3614:A
3607:(
3597::
3581:+
3577:A
3552:A
3528:.
3518:)
3513:+
3509:A
3505:(
3500:=
3495:+
3490:)
3481:A
3477:(
3471:,
3461:+
3457:A
3451:=
3446:+
3441:)
3433:A
3428:(
3422:,
3417:T
3412:)
3407:+
3403:A
3399:(
3394:=
3389:+
3384:)
3379:T
3375:A
3371:(
3358:.
3346:A
3343:=
3338:+
3333:)
3328:+
3324:A
3320:(
3307:.
3293:1
3286:A
3282:=
3277:+
3273:A
3246:A
3233:.
3217:+
3213:A
3188:A
3140:+
3136:A
3132:=
3127:+
3123:A
3119:A
3114:+
3110:A
3089:A
3086:=
3083:A
3078:+
3074:A
3070:A
3043:+
3039:A
3014:A
2979:A
2953:V
2949:V
2946:=
2943:A
2938:+
2934:A
2907:U
2903:U
2900:=
2895:+
2891:A
2887:A
2867:D
2845:+
2841:D
2814:U
2808:+
2804:D
2800:V
2797:=
2792:+
2788:A
2767:D
2747:V
2744:,
2741:U
2715:V
2711:D
2708:U
2705:=
2702:A
2686:.
2674:I
2671:=
2666:+
2662:A
2658:A
2634:.
2629:1
2621:)
2611:A
2607:A
2603:(
2592:A
2588:=
2583:+
2579:A
2554:+
2550:A
2519:A
2515:A
2493:A
2473:A
2462:.
2450:I
2447:=
2444:A
2439:+
2435:A
2410:.
2401:A
2395:1
2387:)
2383:A
2374:A
2369:(
2364:=
2359:+
2355:A
2330:+
2326:A
2301:A
2292:A
2267:A
2243:A
2213:+
2209:A
2184:}
2181:n
2178:,
2175:m
2172:{
2143:A
2121:n
2115:m
2110:K
2102:A
2080:+
2076:A
2050:+
2046:A
2042:=
2037:+
2033:A
2029:)
2026:A
2021:+
2017:A
2013:(
1991:T
1987:A
1966:A
1961:+
1957:A
1936:A
1933:=
1930:A
1927:)
1922:+
1918:A
1914:A
1911:(
1891:A
1869:+
1865:A
1861:A
1839:+
1835:A
1831:A
1811:A
1806:+
1802:A
1781:A
1761:A
1739:+
1735:A
1731:A
1711:A
1689:T
1685:A
1664:A
1659:+
1655:A
1634:A
1629:+
1625:A
1621:=
1616:2
1612:)
1608:A
1603:+
1599:A
1595:(
1573:+
1569:A
1565:A
1562:=
1557:2
1553:)
1547:+
1543:A
1539:A
1536:(
1514:+
1510:A
1506:A
1486:A
1481:+
1477:A
1453:.
1450:A
1445:+
1441:A
1436:=
1426:)
1422:A
1417:+
1413:A
1408:(
1384:A
1379:+
1375:A
1351:.
1346:+
1342:A
1338:A
1334:=
1324:)
1318:+
1314:A
1310:A
1306:(
1276:+
1272:A
1268:A
1245:.
1240:+
1236:A
1231:=
1226:+
1222:A
1218:A
1213:+
1209:A
1180:+
1176:A
1152:.
1149:A
1145:=
1142:A
1137:+
1133:A
1129:A
1101:+
1097:A
1093:A
1064:m
1058:n
1053:K
1043:+
1039:A
1026:A
1010:n
1004:m
999:K
991:A
974:.
958:n
952:n
947:K
937:n
933:I
905:n
899:n
875:n
862:.
848:A
820:)
817:A
814:(
784:A
760:A
746:(
724:)
721:A
718:(
686:n
680:m
675:K
667:A
654:.
640:T
636:A
632:=
623:A
601:R
597:=
593:K
564:A
533:T
529:A
502:n
496:m
491:K
483:A
470:.
454:n
448:m
443:K
416:K
391:n
385:m
360:C
334:R
304:K
264:A
240:A
207:+
203:A
174:A
74:A
45:+
41:A
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