3625:
687:
484:
984:
272:
2124:
Adding up all the ones in a logical matrix may be accomplished in two ways: first summing the rows or first summing the columns. When the row sums are added, the sum is the same as when the column sums are added. In
369:
819:
1841:
1777:
878:
1925:
2110:
2064:
1989:
1113:
1072:
136:
2581:
2066:, it is necessary to use the logical inner product of pairs of logical vectors in rows of this matrix. If this inner product is 0, then the rows are orthogonal. In fact,
2034:
648:
in only two colors can be represented as a (0, 1)-matrix in which the zeros represent pixels of one color and the ones represent pixels of the other color.
3283:
1005:
between two matrices applied component-wise. The complement of a logical matrix is obtained by swapping all zeros and ones for their opposite.
2564:
2013:
Consider the table of group-like structures, where "unneeded" can be denoted 0, and "required" denoted by 1, forming a logical matrix
3497:
2716:
2556:
351:
4 does not hold because when 3 divides 4, there is a remainder of 1. The following set is the set of pairs for which the relation
3588:
2133:
with the rows corresponding to "points" and the columns as "blocks" (generalizing lines made of points). A row sum is called its
479:{\displaystyle {\begin{pmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{pmatrix}},}
2507:
2486:
2429:
2388:
2335:
3507:
3273:
821:. They arise in a variety of representations and have a number of more restricted special forms. They are applied e.g. in
773:
2007:
1930:
A reordering of the rows and columns of such a matrix can assemble all the ones into a rectangular part of the matrix.
709:, that is, the matrix whose entries on the diagonal are all 1, while the others are all 0. More generally, if relation
2651:
2456:
979:{\displaystyle \forall A,B\in U,\quad A\leq B\quad {\text{when}}\quad \forall i,j\quad A_{ij}=1\implies B_{ij}=1.}
3308:
2521:
1044:
is a logical matrix with no columns or rows identically zero. Then the matrix product, using
Boolean arithmetic,
673:
and its variants are matrices that shows which pairs of points are closer than a certain vicinity threshold in a
2148:
with given point degrees and block degrees; or in matrix language, for the existence of a (0, 1)-matrix of type
2855:
2589:
1782:
1718:
1871:
2687:
2085:
2039:
1964:
1088:
1047:
3072:
2709:
565:
531:
267:{\displaystyle m_{i,j}={\begin{cases}1&(x_{i},y_{j})\in R,\\0&(x_{i},y_{j})\not \in R.\end{cases}}}
66:
1712:= 1, it is a column vector. In either case the index equaling 1 is dropped from denotation of the vector.
3147:
2682:
2627:
659:
3303:
2825:
2635:
2372:
1150:
2144:
An early problem in the area was "to find necessary and sufficient conditions for the existence of an
3407:
3278:
3192:
2285:
62:
2677:
164:
3661:
3512:
3402:
3110:
2790:
2246:(1973). "A Fast Expected Time Algorithm for Boolean Matrix Multiplication and Transitive Closure".
1154:
745:
3547:
3476:
3358:
3218:
2815:
2702:
2248:
613:
526:
has ones to indicate incidence between points (or vertices) and lines of a geometry, blocks of a
2628:"II. Feasibility Theorems and Combinatorial Applications §2.12 Matrices composed of 0's and 1's"
3417:
3000:
2805:
2157:
1244:
699:
3363:
3100:
2950:
2945:
2780:
2755:
2750:
1954:
1188:
1173:
3557:
2915:
2745:
2725:
2661:
1863:
1193:
1178:
542:
40:
2469:(2013), "31. Matrices over Finite Fields §31.3 Binary Matrices", in Hogben, Leslie (ed.),
8:
3578:
3552:
3130:
2935:
2925:
2243:
2145:
1626:
594:
569:
502:
is a (0, 1)-matrix, all of whose columns and rows each have exactly one nonzero element.
2016:
3629:
3583:
3573:
3527:
3522:
3451:
3387:
3253:
2990:
2985:
2920:
2910:
2775:
2409:
2294:
2200:
2126:
2113:
1604:
822:
764:
726:
576:
499:
2261:
3666:
3640:
3624:
3427:
3422:
3412:
3392:
3353:
3348:
3177:
3172:
3157:
3152:
3143:
3138:
3085:
2980:
2930:
2875:
2845:
2840:
2820:
2810:
2770:
2647:
2503:
2482:
2452:
2425:
2384:
2356:
2331:
2280:
1135:
663:
3635:
3603:
3532:
3471:
3466:
3446:
3382:
3288:
3258:
3243:
3223:
3162:
3115:
3090:
3080:
3051:
2970:
2965:
2940:
2870:
2850:
2760:
2740:
2639:
2611:
2543:
2474:
2444:
2417:
2376:
2323:
2257:
2190:
2169:
2130:
1200:
1183:
557:
549:
515:
3228:
752:
of the matrix representations of these relations. This product can be computed in
3333:
3268:
3248:
3233:
3213:
3197:
3095:
3026:
3016:
2975:
2860:
2830:
2657:
2577:
2380:
2315:
2185:
2174:
2079:
1334:
1311:
1131:
1082:
994:
703:
691:
670:
641:
634:
580:
523:
82:
54:
3593:
3537:
3517:
3502:
3461:
3338:
3298:
3263:
3187:
3126:
3105:
3046:
3036:
3021:
2955:
2900:
2885:
2795:
2623:
2067:
1469:
1446:
753:
749:
561:
44:
2643:
2602:
Fulkerson, D.R.; Ryser, H.J. (1961). "Widths and heights of (0, 1)-matrices".
2439:
Brualdi, Richard A.; Ryser, Herbert J. (1991). "Combinatorial Matrix Theory".
3655:
3598:
3456:
3397:
3328:
3318:
3313:
3238:
3167:
3041:
3031:
2960:
2880:
2865:
2800:
2466:
2448:
2421:
2360:
2327:
2006:. Relations may be studied by decomposing into concepts, and then noting the
1844:
1648:
869:
598:
538:
519:
1138:; additionally it is a multiplicative lattice due to matrix multiplication.
3481:
3438:
3343:
3056:
2995:
2905:
2785:
2615:
2547:
1424:
1401:
1222:
768:
587:
553:
527:
506:
3323:
3293:
3061:
2895:
2765:
2516:
2495:
2195:
998:
741:
674:
310:
294:
277:
In order to designate the row and column numbers of the matrix, the sets
86:
3374:
2835:
2298:
2267:
2071:
1701:
1560:
1537:
1379:
1356:
1002:
737:
58:
2471:
Handbook of Linear
Algebra (Discrete Mathematics and Its Applications)
3608:
3182:
2180:
1514:
1491:
652:
489:
which includes a diagonal of ones, since each number divides itself.
2283:(December 1948). "Matrix development of the calculus of relations".
2221:
1961:
is called a vector. A particular instance is the universal relation
3542:
2519:(1957). "Combinatorial properties of matrices of zeroes and ones".
2478:
2075:
1289:
1266:
733:
651:
A binary matrix can be used to check the game rules in the game of
763:
Frequently, operations on binary matrices are defined in terms of
686:
2694:
767:
mod 2—that is, the elements are treated as elements of the
359:{(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}.
337:
286:
2156:
with given row and column sums". This problem is solved by the
1582:
347:
4 holds because 2 divides 4 without leaving a remainder, but 3
2534:
Ryser, H.J. (1960). "Traces of matrices of zeroes and ones".
1145:
corresponds to a binary relation. These listed operations on
645:
583:
is a (0, 1)-matrix, and any (0, 1)-matrix arises in this way.
2141:. The sum of point degrees equals the sum of block degrees.
260:
662:
of two bits, transformed by 2x2 logical matrices, forms a
363:
The corresponding representation as a logical matrix is
378:
72:
2088:
2042:
2019:
1967:
1874:
1785:
1721:
1091:
1050:
881:
776:
372:
139:
2241:
836:binary matrices is equal to 2, and is thus finite.
118:whose row and column indices index the elements of
2354:
2104:
2058:
2028:
1983:
1919:
1835:
1771:
1107:
1066:
978:
814:{\displaystyle {\mathbf {GF}}(2)=\mathbb {Z} _{2}}
813:
478:
266:
2622:
1126:As a mathematical structure, the Boolean algebra
3653:
2441:Encyclopedia of Mathematics and its Applications
2414:Encyclopedia of Mathematics and its Applications
2369:Encyclopedia of Mathematics and its Applications
633:th number. This representation is useful in the
2266:— The algorithm relies on addition being
2601:
1998:, a maximal rectangular relation contained in
2710:
2565:Bulletin of the American Mathematical Society
2416:. Vol. 108. Cambridge University Press.
2363:(1999). "I. Examples and basic definitions".
1941:is an arbitrary logical vector, the relation
1153:, where the matrix multiplication represents
690:Multiplication of two logical matrices using
2443:. Vol. 39. Cambridge University Press.
2438:
568:, and a 1 on the diagonal corresponds to a
3284:Fundamental (linear differential equation)
2717:
2703:
2322:. Cambridge University Press. p. 91.
2279:
956:
952:
545:is a (0, 1)-matrix with constant row sums.
509:is a special case of a permutation matrix.
2576:
1836:{\displaystyle (Q_{j}),\,j=1,2,\ldots ,n}
1805:
1772:{\displaystyle (P_{i}),\,i=1,2,\ldots ,m}
1741:
801:
343:evenly, with no remainder. For example, 2
126:, respectively, such that the entries of
114:can be represented by the logical matrix
53:Such a matrix can be used to represent a
2473:(2nd ed.), Chapman & Hall/CRC,
2412:(2006). "Combinatorial Matrix Classes".
2219:
1920:{\displaystyle m_{ij}=P_{i}\land Q_{j}.}
1708:= 1, the vector is a row vector, and if
685:
3589:Matrix representation of conic sections
2408:
2314:
2105:{\displaystyle RR^{\operatorname {T} }}
2059:{\displaystyle RR^{\operatorname {T} }}
1984:{\displaystyle hh^{\operatorname {T} }}
1108:{\displaystyle AA^{\operatorname {T} }}
1067:{\displaystyle A^{\operatorname {T} }A}
3654:
2500:Boolean Matrix Theory and Applications
2119:
2698:
2554:
2533:
2515:
2465:
2349:
2347:
732:If the Boolean domain is viewed as a
2318:(2013). "6: Relations and Vectors".
2310:
2308:
2220:Petersen, Kjeld (February 8, 2013).
305:ranges from 1 to the cardinality of
2582:"Zero-one matrices with zero trace"
2494:
1937:be the vector of all ones. Then if
744:, the matrix representation of the
73:Matrix representation of a relation
13:
2724:
2344:
2129:, the matrix is interpreted as an
2097:
2082:. Consequently there are zeros in
2051:
1976:
1160:
1100:
1056:
920:
882:
681:
548:A logical matrix may represent an
14:
3678:
2670:
2305:
748:of two relations is equal to the
698:The matrix representation of the
564:, symmetric matrices to ordinary
492:
3623:
1949:has constant rows determined by
1149:, and ordering, correspond to a
782:
779:
736:, where addition corresponds to
612:) (0, 1)-matrix, where π is the
3491:Used in science and engineering
2604:Canadian Journal of Mathematics
2536:Canadian Journal of Mathematics
2522:Canadian Journal of Mathematics
2371:. Vol. 69 (2nd ed.).
932:
919:
913:
903:
604:numbers can be described as an
2734:Explicitly constrained entries
2590:Pacific Journal of Mathematics
2273:
2235:
2213:
1799:
1786:
1735:
1722:
953:
856:denote the set of all logical
793:
787:
245:
219:
198:
172:
1:
3508:Fundamental (computer vision)
2401:
2262:10.1016/s0019-9958(73)90228-3
1843:are two logical vectors. The
61:. It is an important tool in
16:Matrix of binary truth values
2557:"Matrices of Zeros and Ones"
2381:10.1017/CBO9780511549533.001
1559:Commutative-and-associative
572:at the corresponding vertex.
67:theoretical computer science
7:
3274:Duplication and elimination
3073:eigenvalues or eigenvectors
2683:Encyclopedia of Mathematics
2626:; Fulkerson, D.R. (2016) .
2163:
10:
3683:
3207:With specific applications
2836:Discrete Fourier Transform
2636:Princeton University Press
2373:Cambridge University Press
2177:(a binary De Bruijn torus)
2137:, and a column sum is the
839:
316:
285:are indexed with positive
3617:
3566:
3498:Cabibbo–Kobayashi–Maskawa
3490:
3436:
3372:
3206:
3125:Satisfying conditions on
3124:
3070:
3009:
2733:
2644:10.1515/9781400875184-004
2286:Journal of Symbolic Logic
2036:To calculate elements of
63:combinatorial mathematics
2449:10.1017/CBO9781107325708
2422:10.1017/CBO9780511721182
2328:10.1017/CBO9780511778810
2206:
1155:composition of relations
1141:Every logical matrix in
625:is 1 if and only if the
579:of a simple, undirected
2856:Generalized permutation
2249:Information and Control
2112:, and it fails to be a
2008:induced concept lattice
2002:is called a concept in
828:The number of distinct
702:on a finite set is the
614:prime-counting function
560:matrices correspond to
3630:Mathematics portal
2616:10.4153/CJM-1961-020-3
2548:10.4153/CJM-1960-040-0
2320:Relational Mathematics
2106:
2060:
2030:
1985:
1921:
1837:
1773:
1166:Group-like structures
1109:
1068:
980:
815:
740:and multiplication to
695:
480:
268:
43:with entries from the
2270:, cf. p.134 (bottom).
2107:
2061:
2031:
1994:For a given relation
1986:
1955:calculus of relations
1922:
1838:
1774:
1679:equals one, then the
1151:calculus of relations
1110:
1069:
1008:Every logical matrix
981:
816:
689:
629:th prime divides the
481:
333:holds if and only if
309:. See the article on
293:ranges from 1 to the
269:
2555:Ryser, H.J. (1960).
2244:O'Neil, Elizabeth J.
2242:O'Neil, Patrick E.;
2086:
2040:
2017:
1965:
1872:
1864:rectangular relation
1783:
1719:
1089:
1048:
997:with the operations
879:
774:
637:factoring algorithm.
543:analysis of variance
370:
321:The binary relation
137:
3579:Linear independence
2826:Diagonally dominant
2410:Brualdi, Richard A.
2146:incidence structure
2120:Row and column sums
1167:
329:is defined so that
85:between the finite
3584:Matrix exponential
3574:Jordan normal form
3408:Fisher information
3279:Euclidean distance
3193:Totally unimodular
2638:. pp. 79–91.
2357:Jungnickel, Dieter
2281:Copilowish, Irving
2201:Three-valued logic
2158:Gale–Ryser theorem
2127:incidence geometry
2114:universal relation
2102:
2056:
2029:{\displaystyle R.}
2026:
1981:
1917:
1833:
1769:
1605:Commutative monoid
1165:
1105:
1085:, and the product
1064:
976:
823:XOR-satisfiability
811:
765:modular arithmetic
727:reflexive relation
696:
577:biadjacency matrix
500:permutation matrix
476:
467:
264:
259:
57:between a pair of
3649:
3648:
3641:Category:Matrices
3513:Fuzzy associative
3403:Doubly stochastic
3111:Positive-definite
2791:Block tridiagonal
2632:Flows in Networks
2509:978-0-8247-1788-9
2488:978-0-429-18553-3
2431:978-0-521-86565-4
2390:978-0-521-44432-3
2337:978-0-511-77881-0
2078:is orthogonal to
2070:is orthogonal to
1669:
1668:
917:
852:be given and let
700:equality relation
664:transition system
660:four valued logic
313:for more detail.
3674:
3636:List of matrices
3628:
3627:
3604:Row echelon form
3548:State transition
3477:Seidel adjacency
3359:Totally positive
3219:Alternating sign
2816:Complex Hadamard
2719:
2712:
2705:
2696:
2695:
2691:
2678:"Logical matrix"
2665:
2619:
2598:
2586:
2573:
2561:
2551:
2530:
2512:
2491:
2462:
2435:
2395:
2394:
2351:
2342:
2341:
2316:Schmidt, Gunther
2312:
2303:
2302:
2277:
2271:
2265:
2239:
2233:
2232:
2230:
2228:
2217:
2191:Redheffer matrix
2170:List of matrices
2131:incidence matrix
2111:
2109:
2108:
2103:
2101:
2100:
2065:
2063:
2062:
2057:
2055:
2054:
2035:
2033:
2032:
2027:
1990:
1988:
1987:
1982:
1980:
1979:
1926:
1924:
1923:
1918:
1913:
1912:
1900:
1899:
1887:
1886:
1842:
1840:
1839:
1834:
1798:
1797:
1778:
1776:
1775:
1770:
1734:
1733:
1687:logical matrix (
1168:
1164:
1114:
1112:
1111:
1106:
1104:
1103:
1073:
1071:
1070:
1065:
1060:
1059:
1039:
1024:has a transpose
1023:
985:
983:
982:
977:
969:
968:
945:
944:
918:
915:
820:
818:
817:
812:
810:
809:
804:
786:
785:
720:
550:adjacency matrix
530:, or edges of a
516:incidence matrix
485:
483:
482:
477:
472:
471:
328:
273:
271:
270:
265:
263:
262:
244:
243:
231:
230:
197:
196:
184:
183:
155:
154:
109:
52:
3682:
3681:
3677:
3676:
3675:
3673:
3672:
3671:
3662:Boolean algebra
3652:
3651:
3650:
3645:
3622:
3613:
3562:
3486:
3432:
3368:
3202:
3120:
3066:
3005:
2806:Centrosymmetric
2729:
2723:
2676:
2673:
2668:
2654:
2584:
2578:Fulkerson, D.R.
2559:
2510:
2489:
2459:
2432:
2404:
2399:
2398:
2391:
2352:
2345:
2338:
2313:
2306:
2278:
2274:
2240:
2236:
2226:
2224:
2218:
2214:
2209:
2186:Disjunct matrix
2166:
2122:
2096:
2092:
2087:
2084:
2083:
2050:
2046:
2041:
2038:
2037:
2018:
2015:
2014:
1975:
1971:
1966:
1963:
1962:
1908:
1904:
1895:
1891:
1879:
1875:
1873:
1870:
1869:
1793:
1789:
1784:
1781:
1780:
1729:
1725:
1720:
1717:
1716:
1695:
1163:
1161:Logical vectors
1099:
1095:
1090:
1087:
1086:
1083:identity matrix
1055:
1051:
1049:
1046:
1045:
1037:
1025:
1021:
1009:
995:Boolean algebra
961:
957:
937:
933:
914:
880:
877:
876:
864:matrices. Then
842:
805:
800:
799:
778:
777:
775:
772:
771:
714:
704:identity matrix
692:Boolean algebra
684:
682:Some properties
671:recurrence plot
635:quadratic sieve
624:
581:bipartite graph
562:directed graphs
524:finite geometry
495:
466:
465:
460:
455:
450:
444:
443:
438:
433:
428:
422:
421:
416:
411:
406:
400:
399:
394:
389:
384:
374:
373:
371:
368:
367:
326:
319:
258:
257:
239:
235:
226:
222:
217:
211:
210:
192:
188:
179:
175:
170:
160:
159:
144:
140:
138:
135:
134:
130:are defined by
97:
83:binary relation
75:
55:binary relation
47:
29:relation matrix
17:
12:
11:
5:
3680:
3670:
3669:
3664:
3647:
3646:
3644:
3643:
3638:
3633:
3618:
3615:
3614:
3612:
3611:
3606:
3601:
3596:
3594:Perfect matrix
3591:
3586:
3581:
3576:
3570:
3568:
3564:
3563:
3561:
3560:
3555:
3550:
3545:
3540:
3535:
3530:
3525:
3520:
3515:
3510:
3505:
3500:
3494:
3492:
3488:
3487:
3485:
3484:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3443:
3441:
3434:
3433:
3431:
3430:
3425:
3420:
3415:
3410:
3405:
3400:
3395:
3390:
3385:
3379:
3377:
3370:
3369:
3367:
3366:
3364:Transformation
3361:
3356:
3351:
3346:
3341:
3336:
3331:
3326:
3321:
3316:
3311:
3306:
3301:
3296:
3291:
3286:
3281:
3276:
3271:
3266:
3261:
3256:
3251:
3246:
3241:
3236:
3231:
3226:
3221:
3216:
3210:
3208:
3204:
3203:
3201:
3200:
3195:
3190:
3185:
3180:
3175:
3170:
3165:
3160:
3155:
3150:
3141:
3135:
3133:
3122:
3121:
3119:
3118:
3113:
3108:
3103:
3101:Diagonalizable
3098:
3093:
3088:
3083:
3077:
3075:
3071:Conditions on
3068:
3067:
3065:
3064:
3059:
3054:
3049:
3044:
3039:
3034:
3029:
3024:
3019:
3013:
3011:
3007:
3006:
3004:
3003:
2998:
2993:
2988:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2951:Skew-symmetric
2948:
2946:Skew-Hermitian
2943:
2938:
2933:
2928:
2923:
2918:
2913:
2908:
2903:
2898:
2893:
2888:
2883:
2878:
2873:
2868:
2863:
2858:
2853:
2848:
2843:
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2788:
2783:
2781:Block-diagonal
2778:
2773:
2768:
2763:
2758:
2756:Anti-symmetric
2753:
2751:Anti-Hermitian
2748:
2743:
2737:
2735:
2731:
2730:
2722:
2721:
2714:
2707:
2699:
2693:
2692:
2672:
2671:External links
2669:
2667:
2666:
2652:
2624:Ford Jr., L.R.
2620:
2599:
2574:
2552:
2531:
2513:
2508:
2492:
2487:
2479:10.1201/b16113
2463:
2457:
2436:
2430:
2405:
2403:
2400:
2397:
2396:
2389:
2375:. p. 18.
2361:Lenz, Hanfried
2355:Beth, Thomas;
2343:
2336:
2304:
2293:(4): 193–203.
2272:
2234:
2211:
2210:
2208:
2205:
2204:
2203:
2198:
2193:
2188:
2183:
2178:
2172:
2165:
2162:
2121:
2118:
2099:
2095:
2091:
2068:small category
2053:
2049:
2045:
2025:
2022:
1978:
1974:
1970:
1928:
1927:
1916:
1911:
1907:
1903:
1898:
1894:
1890:
1885:
1882:
1878:
1855:results in an
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1804:
1801:
1796:
1792:
1788:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1740:
1737:
1732:
1728:
1724:
1698:logical vector
1691:
1667:
1666:
1663:
1660:
1657:
1654:
1651:
1645:
1644:
1641:
1638:
1635:
1632:
1629:
1623:
1622:
1619:
1616:
1613:
1610:
1607:
1601:
1600:
1597:
1594:
1591:
1588:
1585:
1579:
1578:
1575:
1572:
1569:
1566:
1563:
1556:
1555:
1552:
1549:
1546:
1543:
1540:
1533:
1532:
1529:
1526:
1523:
1520:
1517:
1510:
1509:
1506:
1503:
1500:
1497:
1494:
1488:
1487:
1484:
1481:
1478:
1475:
1472:
1465:
1464:
1461:
1458:
1455:
1452:
1449:
1443:
1442:
1439:
1436:
1433:
1430:
1427:
1420:
1419:
1416:
1413:
1410:
1407:
1404:
1398:
1397:
1394:
1391:
1388:
1385:
1382:
1375:
1374:
1371:
1368:
1365:
1362:
1359:
1353:
1352:
1349:
1346:
1343:
1340:
1337:
1330:
1329:
1326:
1323:
1320:
1317:
1314:
1308:
1307:
1304:
1301:
1298:
1295:
1292:
1285:
1284:
1281:
1278:
1275:
1272:
1269:
1263:
1262:
1259:
1256:
1253:
1250:
1247:
1245:Small category
1241:
1240:
1237:
1234:
1231:
1228:
1225:
1219:
1218:
1215:
1212:
1209:
1206:
1203:
1197:
1196:
1191:
1186:
1181:
1176:
1171:
1162:
1159:
1102:
1098:
1094:
1063:
1058:
1054:
1033:
1017:
987:
986:
975:
972:
967:
964:
960:
955:
951:
948:
943:
940:
936:
931:
928:
925:
922:
912:
909:
906:
902:
899:
896:
893:
890:
887:
884:
841:
838:
808:
803:
798:
795:
792:
789:
784:
781:
750:matrix product
683:
680:
679:
678:
667:
656:
649:
638:
620:
584:
573:
546:
535:
512:
511:
510:
494:
493:Other examples
491:
487:
486:
475:
470:
464:
461:
459:
456:
454:
451:
449:
446:
445:
442:
439:
437:
434:
432:
429:
427:
424:
423:
420:
417:
415:
412:
410:
407:
405:
402:
401:
398:
395:
393:
390:
388:
385:
383:
380:
379:
377:
361:
360:
318:
315:
275:
274:
261:
256:
253:
250:
247:
242:
238:
234:
229:
225:
221:
218:
216:
213:
212:
209:
206:
203:
200:
195:
191:
187:
182:
178:
174:
171:
169:
166:
165:
163:
158:
153:
150:
147:
143:
74:
71:
45:Boolean domain
33:Boolean matrix
21:logical matrix
15:
9:
6:
4:
3:
2:
3679:
3668:
3665:
3663:
3660:
3659:
3657:
3642:
3639:
3637:
3634:
3632:
3631:
3626:
3620:
3619:
3616:
3610:
3607:
3605:
3602:
3600:
3599:Pseudoinverse
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3571:
3569:
3567:Related terms
3565:
3559:
3558:Z (chemistry)
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3495:
3493:
3489:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3444:
3442:
3440:
3435:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3380:
3378:
3376:
3371:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3295:
3292:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3211:
3209:
3205:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3145:
3142:
3140:
3137:
3136:
3134:
3132:
3128:
3123:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3078:
3076:
3074:
3069:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3014:
3012:
3008:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2922:
2919:
2917:
2916:Pentadiagonal
2914:
2912:
2909:
2907:
2904:
2902:
2899:
2897:
2894:
2892:
2889:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2746:Anti-diagonal
2744:
2742:
2739:
2738:
2736:
2732:
2727:
2720:
2715:
2713:
2708:
2706:
2701:
2700:
2697:
2689:
2685:
2684:
2679:
2675:
2674:
2663:
2659:
2655:
2653:9781400875184
2649:
2645:
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2609:
2605:
2600:
2596:
2592:
2591:
2583:
2579:
2575:
2571:
2567:
2566:
2558:
2553:
2549:
2545:
2541:
2537:
2532:
2528:
2524:
2523:
2518:
2514:
2511:
2505:
2501:
2497:
2493:
2490:
2484:
2480:
2476:
2472:
2468:
2464:
2460:
2458:0-521-32265-0
2454:
2450:
2446:
2442:
2437:
2433:
2427:
2423:
2419:
2415:
2411:
2407:
2406:
2392:
2386:
2382:
2378:
2374:
2370:
2366:
2365:Design Theory
2362:
2358:
2350:
2348:
2339:
2333:
2329:
2325:
2321:
2317:
2311:
2309:
2300:
2296:
2292:
2288:
2287:
2282:
2276:
2269:
2263:
2259:
2255:
2251:
2250:
2245:
2238:
2223:
2216:
2212:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2176:
2173:
2171:
2168:
2167:
2161:
2159:
2155:
2152: ×
2151:
2147:
2142:
2140:
2136:
2132:
2128:
2117:
2115:
2093:
2089:
2081:
2077:
2073:
2069:
2047:
2043:
2023:
2020:
2011:
2009:
2005:
2001:
1997:
1992:
1972:
1968:
1960:
1956:
1952:
1948:
1944:
1940:
1936:
1931:
1914:
1909:
1905:
1901:
1896:
1892:
1888:
1883:
1880:
1876:
1868:
1867:
1866:
1865:
1862:
1858:
1854:
1850:
1846:
1845:outer product
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1802:
1794:
1790:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1738:
1730:
1726:
1713:
1711:
1707:
1703:
1699:
1694:
1690:
1686:
1682:
1678:
1674:
1664:
1661:
1658:
1655:
1652:
1650:
1649:Abelian group
1647:
1646:
1642:
1639:
1636:
1633:
1630:
1628:
1625:
1624:
1620:
1617:
1614:
1611:
1608:
1606:
1603:
1602:
1598:
1595:
1592:
1589:
1586:
1584:
1581:
1580:
1576:
1573:
1570:
1567:
1564:
1562:
1558:
1557:
1553:
1550:
1547:
1544:
1541:
1539:
1535:
1534:
1530:
1527:
1524:
1521:
1518:
1516:
1512:
1511:
1507:
1504:
1501:
1498:
1495:
1493:
1490:
1489:
1485:
1482:
1479:
1476:
1473:
1471:
1467:
1466:
1462:
1459:
1456:
1453:
1450:
1448:
1445:
1444:
1440:
1437:
1434:
1431:
1428:
1426:
1422:
1421:
1417:
1414:
1411:
1408:
1405:
1403:
1400:
1399:
1395:
1392:
1389:
1386:
1383:
1381:
1377:
1376:
1372:
1369:
1366:
1363:
1360:
1358:
1355:
1354:
1350:
1347:
1344:
1341:
1338:
1336:
1332:
1331:
1327:
1324:
1321:
1318:
1315:
1313:
1310:
1309:
1305:
1302:
1299:
1296:
1293:
1291:
1287:
1286:
1282:
1279:
1276:
1273:
1270:
1268:
1265:
1264:
1260:
1257:
1254:
1251:
1248:
1246:
1243:
1242:
1238:
1235:
1232:
1229:
1226:
1224:
1221:
1220:
1216:
1213:
1210:
1207:
1204:
1202:
1201:Partial magma
1199:
1198:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1169:
1158:
1156:
1152:
1148:
1144:
1139:
1137:
1133:
1129:
1124:
1122:
1118:
1115:contains the
1096:
1092:
1084:
1081:
1077:
1074:contains the
1061:
1052:
1043:
1036:
1032:
1028:
1020:
1016:
1012:
1006:
1004:
1000:
996:
992:
973:
970:
965:
962:
958:
949:
946:
941:
938:
934:
929:
926:
923:
910:
907:
904:
900:
897:
894:
891:
888:
885:
875:
874:
873:
871:
870:partial order
867:
863:
859:
855:
851:
847:
837:
835:
831:
826:
824:
806:
796:
790:
770:
766:
761:
759:
755:
751:
747:
743:
739:
735:
730:
728:
724:
718:
712:
708:
705:
701:
693:
688:
676:
672:
668:
665:
661:
657:
654:
650:
647:
643:
639:
636:
632:
628:
623:
619:
615:
611:
607:
603:
601:
596:
593:
590:of a list of
589:
588:prime factors
585:
582:
578:
574:
571:
567:
563:
559:
555:
551:
547:
544:
540:
539:design matrix
536:
533:
529:
525:
521:
520:combinatorics
517:
513:
508:
504:
503:
501:
497:
496:
490:
473:
468:
462:
457:
452:
447:
440:
435:
430:
425:
418:
413:
408:
403:
396:
391:
386:
381:
375:
366:
365:
364:
358:
357:
356:
354:
350:
346:
342:
339:
336:
332:
324:
314:
312:
308:
304:
300:
296:
292:
288:
284:
280:
254:
251:
248:
240:
236:
232:
227:
223:
214:
207:
204:
201:
193:
189:
185:
180:
176:
167:
161:
156:
151:
148:
145:
141:
133:
132:
131:
129:
125:
121:
117:
113:
108:
104:
100:
95:
91:
88:
84:
80:
70:
68:
64:
60:
56:
50:
46:
42:
38:
37:(0, 1)-matrix
34:
30:
26:
25:binary matrix
22:
3621:
3553:Substitution
3439:graph theory
2936:Quaternionic
2926:Persymmetric
2890:
2681:
2631:
2607:
2603:
2594:
2588:
2569:
2563:
2539:
2535:
2526:
2520:
2499:
2496:Kim, Ki Hang
2470:
2440:
2413:
2368:
2364:
2319:
2290:
2284:
2275:
2256:(2): 132–8.
2253:
2247:
2237:
2225:. Retrieved
2215:
2153:
2149:
2143:
2139:block degree
2138:
2135:point degree
2134:
2123:
2012:
2003:
1999:
1995:
1993:
1958:
1950:
1946:
1942:
1938:
1934:
1932:
1929:
1860:
1856:
1852:
1848:
1714:
1709:
1705:
1697:
1692:
1688:
1684:
1680:
1676:
1672:
1670:
1536:Associative
1513:Commutative
1468:Commutative
1425:unital magma
1423:Commutative
1402:Unital magma
1378:Commutative
1333:Commutative
1288:Commutative
1223:Semigroupoid
1189:Cancellation
1146:
1142:
1140:
1127:
1125:
1120:
1116:
1079:
1075:
1041:
1034:
1030:
1026:
1018:
1014:
1010:
1007:
990:
988:
865:
861:
857:
853:
849:
845:
843:
833:
829:
827:
769:Galois field
762:
757:
731:
722:
716:
710:
706:
697:
642:bitmap image
630:
626:
621:
617:
609:
605:
599:
591:
554:graph theory
528:block design
507:Costas array
488:
362:
352:
348:
344:
340:
334:
330:
327:{1, 2, 3, 4}
322:
320:
311:indexed sets
306:
302:
298:
290:
282:
278:
276:
127:
123:
119:
115:
111:
106:
102:
98:
93:
89:
87:indexed sets
78:
76:
48:
36:
32:
28:
24:
20:
18:
3528:Hamiltonian
3452:Biadjacency
3388:Correlation
3304:Householder
3254:Commutation
2991:Vandermonde
2986:Tridiagonal
2921:Permutation
2911:Nonnegative
2896:Matrix unit
2776:Bisymmetric
2610:: 239–255.
2542:: 463–476.
2517:Ryser, H.J.
2467:Botha, J.D.
2222:"Binmatrix"
2196:Truth table
1194:Commutative
1179:Associative
1134:ordered by
746:composition
742:logical AND
675:phase space
644:containing
595:square-free
325:on the set
295:cardinality
59:finite sets
3656:Categories
3428:Transition
3423:Stochastic
3393:Covariance
3375:statistics
3354:Symplectic
3349:Similarity
3178:Unimodular
3173:Orthogonal
3158:Involutory
3153:Invertible
3148:Projection
3144:Idempotent
3086:Convergent
2981:Triangular
2931:Polynomial
2876:Hessenberg
2846:Equivalent
2841:Elementary
2821:Copositive
2811:Conference
2771:Bidiagonal
2572:: 442–464.
2402:References
2353:E.g., see
2268:idempotent
2227:August 11,
2072:quasigroup
1702:bit string
1561:quasigroup
1538:quasigroup
1380:quasigroup
1357:Quasigroup
1123:identity.
738:logical OR
713:satisfies
297:(size) of
3609:Wronskian
3533:Irregular
3523:Gell-Mann
3472:Laplacian
3467:Incidence
3447:Adjacency
3418:Precision
3383:Centering
3289:Generator
3259:Confusion
3244:Circulant
3224:Augmented
3183:Unipotent
3163:Nilpotent
3139:Congruent
3116:Stieltjes
3091:Defective
3081:Companion
3052:Redheffer
2971:Symmetric
2966:Sylvester
2941:Signature
2871:Hermitian
2851:Frobenius
2761:Arrowhead
2741:Alternant
2688:EMS Press
2181:Bit array
2175:Binatorix
1953:. In the
1902:∧
1825:…
1761:…
1665:Required
1643:Unneeded
1621:Required
1599:Unneeded
1577:Required
1554:Unneeded
1531:Required
1515:semigroup
1508:Unneeded
1492:Semigroup
1486:Required
1463:Unneeded
1441:Required
1418:Unneeded
1396:Required
1373:Unneeded
1351:Required
1328:Unneeded
1306:Required
1283:Unneeded
1261:Unneeded
1239:Unneeded
1217:Unneeded
1136:inclusion
989:In fact,
954:⟹
921:∀
908:≤
895:∈
883:∀
872:given by
558:symmetric
202:∈
51:= {0, 1}.
3667:Matrices
3437:Used in
3373:Used in
3334:Rotation
3309:Jacobian
3269:Distance
3249:Cofactor
3234:Carleman
3214:Adjugate
3198:Weighing
3131:inverses
3127:products
3096:Definite
3027:Identity
3017:Exchange
3010:Constant
2976:Toeplitz
2861:Hadamard
2831:Diagonal
2597:: 831–6.
2580:(1960).
2529:: 371–7.
2498:(1982),
2164:See also
2076:groupoid
1957:such an
1715:Suppose
1662:Required
1659:Required
1656:Required
1653:Required
1640:Required
1637:Required
1634:Required
1631:Required
1618:Unneeded
1615:Required
1612:Required
1609:Required
1596:Unneeded
1593:Required
1590:Required
1587:Required
1574:Required
1571:Unneeded
1568:Required
1565:Required
1551:Required
1548:Unneeded
1545:Required
1542:Required
1528:Unneeded
1525:Unneeded
1522:Required
1519:Required
1505:Unneeded
1502:Unneeded
1499:Required
1496:Required
1483:Required
1480:Required
1477:Unneeded
1474:Required
1460:Required
1457:Required
1454:Unneeded
1451:Required
1438:Unneeded
1435:Required
1432:Unneeded
1429:Required
1415:Unneeded
1412:Required
1409:Unneeded
1406:Required
1393:Required
1390:Unneeded
1387:Unneeded
1384:Required
1370:Required
1367:Unneeded
1364:Unneeded
1361:Required
1348:Unneeded
1345:Unneeded
1342:Unneeded
1339:Required
1325:Unneeded
1322:Unneeded
1319:Unneeded
1316:Required
1303:Required
1300:Required
1297:Required
1294:Unneeded
1290:Groupoid
1280:Required
1277:Required
1274:Required
1271:Unneeded
1267:Groupoid
1258:Unneeded
1255:Required
1252:Required
1249:Unneeded
1236:Unneeded
1233:Unneeded
1230:Required
1227:Unneeded
1214:Unneeded
1211:Unneeded
1208:Unneeded
1205:Unneeded
1184:Identity
1130:forms a
1040:Suppose
993:forms a
754:expected
734:semiring
355:holds.
287:integers
249:∉
110:), then
3538:Overlap
3503:Density
3462:Edmonds
3339:Seifert
3299:Hessian
3264:Coxeter
3188:Unitary
3106:Hurwitz
3037:Of ones
3022:Hilbert
2956:Skyline
2901:Metzler
2891:Logical
2886:Integer
2796:Boolean
2728:classes
2690:, 2001
2662:0159700
2299:2267134
1696:) is a
1174:Closure
1132:lattice
840:Lattice
756:time O(
602:-smooth
338:divides
317:Example
3457:Degree
3398:Design
3329:Random
3319:Payoff
3314:Moment
3239:Cartan
3229:Bézout
3168:Normal
3042:Pascal
3032:Lehmer
2961:Sparse
2881:Hollow
2866:Hankel
2801:Cauchy
2726:Matrix
2660:
2650:
2506:
2485:
2455:
2428:
2387:
2334:
2297:
2074:, and
1583:Monoid
1001:&
868:has a
646:pixels
616:, and
566:graphs
556:: non-
301:, and
41:matrix
3518:Gamma
3482:Tutte
3344:Shear
3057:Shift
3047:Pauli
2996:Walsh
2906:Moore
2786:Block
2585:(PDF)
2560:(PDF)
2295:JSTOR
2207:Notes
2080:magma
1704:. If
1627:Group
1335:magma
1312:Magma
725:is a
721:then
532:graph
81:is a
39:is a
35:, or
3324:Pick
3294:Gram
3062:Zero
2766:Band
2648:ISBN
2504:ISBN
2483:ISBN
2453:ISBN
2426:ISBN
2385:ISBN
2332:ISBN
2229:2017
1933:Let
1851:and
1779:and
1470:loop
1447:Loop
916:when
848:and
844:Let
832:-by-
715:I ⊆
658:The
608:× π(
586:The
575:The
570:loop
522:and
281:and
122:and
96:(so
92:and
65:and
3413:Hat
3146:or
3129:or
2640:doi
2612:doi
2544:doi
2475:doi
2445:doi
2418:doi
2377:doi
2324:doi
2258:doi
1947:v h
1847:of
1700:or
1675:or
1671:If
1029:= (
1013:= (
999:and
760:).
552:in
541:in
518:in
514:An
331:aRb
77:If
3658::
2686:,
2680:,
2658:MR
2656:.
2646:.
2634:.
2630:.
2608:13
2606:.
2595:10
2593:.
2587:.
2570:66
2568:.
2562:.
2540:12
2538:.
2525:.
2502:,
2481:,
2451:.
2424:.
2383:.
2367:.
2359:;
2346:^
2330:.
2307:^
2291:13
2289:.
2254:22
2252:.
2160:.
2116:.
2010:.
1991:.
1945:=
1859:×
1693:ij
1683:×
1157:.
1119:×
1078:×
1038:).
1035:ji
1019:ij
1003:or
974:1.
860:×
825:.
729:.
669:A
653:Go
640:A
622:ij
597:,
537:A
505:A
498:A
289::
101:⊆
69:.
31:,
27:,
23:,
19:A
3543:S
3001:Z
2718:e
2711:t
2704:v
2664:.
2642::
2618:.
2614::
2550:.
2546::
2527:9
2477::
2461:.
2447::
2434:.
2420::
2393:.
2379::
2340:.
2326::
2301:.
2264:.
2260::
2231:.
2154:b
2150:v
2098:T
2094:R
2090:R
2052:T
2048:R
2044:R
2024:.
2021:R
2004:R
2000:R
1996:R
1977:T
1973:h
1969:h
1959:R
1951:v
1943:R
1939:v
1935:h
1915:.
1910:j
1906:Q
1897:i
1893:P
1889:=
1884:j
1881:i
1877:m
1861:n
1857:m
1853:Q
1849:P
1831:n
1828:,
1822:,
1819:2
1816:,
1813:1
1810:=
1807:j
1803:,
1800:)
1795:j
1791:Q
1787:(
1767:m
1764:,
1758:,
1755:2
1752:,
1749:1
1746:=
1743:i
1739:,
1736:)
1731:i
1727:P
1723:(
1710:n
1706:m
1689:m
1685:n
1681:m
1677:n
1673:m
1147:U
1143:U
1128:U
1121:n
1117:n
1101:T
1097:A
1093:A
1080:m
1076:m
1062:A
1057:T
1053:A
1042:A
1031:A
1027:A
1022:)
1015:A
1011:A
991:U
971:=
966:j
963:i
959:B
950:1
947:=
942:j
939:i
935:A
930:j
927:,
924:i
911:B
905:A
901:,
898:U
892:B
889:,
886:A
866:U
862:n
858:m
854:U
850:m
846:n
834:n
830:m
807:2
802:Z
797:=
794:)
791:2
788:(
783:F
780:G
758:n
723:R
719:,
717:R
711:R
707:I
694:.
677:.
666:.
655:.
631:i
627:j
618:a
610:n
606:m
600:n
592:m
534:.
474:,
469:)
463:1
458:0
453:0
448:0
441:0
436:1
431:0
426:0
419:1
414:0
409:1
404:0
397:1
392:1
387:1
382:1
376:(
353:R
349:R
345:R
341:b
335:a
323:R
307:Y
303:j
299:X
291:i
283:Y
279:X
255:.
252:R
246:)
241:j
237:y
233:,
228:i
224:x
220:(
215:0
208:,
205:R
199:)
194:j
190:y
186:,
181:i
177:x
173:(
168:1
162:{
157:=
152:j
149:,
146:i
142:m
128:M
124:Y
120:X
116:M
112:R
107:Y
105:×
103:X
99:R
94:Y
90:X
79:R
49:B
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