2877:
1582:(a positive integer), the column sums are equal to 1, and all cells are non-negative (the sum of the row sums being equal to the number of columns). Any matrix in this form can be expressed as a convex combination of matrices in the same form made up of 0s and 1s. The proof is to replace the
1352:
1614:
In the same way it is possible to replicate columns as well as rows, but the result of recombination is not necessarily limited to 0s and 1s. A different generalisation (with a significantly harder proof) has been put forward by R. M. Caron et al.
1128:
1214:
799:
The product of two doubly stochastic matrices is doubly stochastic. However, the inverse of a nonsingular doubly stochastic matrix need not be doubly stochastic (indeed, the inverse is doubly stochastic if it has nonnegative
638:
420:
because one of these constraints is dependent, as the sum of the row sums must equal the sum of the column sums.) Moreover, the entries are all constrained to be non-negative and less than or equal to 1.
762:
1027:
by removing scalar multiples of permutation matrices until we arrive at the zero matrix, at which point we will have constructed a convex combination of permutation matrices equal to the original
175:
191:: if every row sums to 1 then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal.
1252:
1247:
684:
1037:
93:
492:
306:
229:
930:
526:
462:
333:
263:
395:
366:
958:
844:
1133:
960:. Proofs of this conjecture were published in 1980 by B. Gyires and in 1981 by G. P. Egorychev and D. I. Falikman; for this work, Egorychev and Falikman won the
418:
872:
546:
551:
2535:
17:
2749:
1968:
692:
2840:
1712:, IMS Lecture Notes â Monographs Series, edited by L. RĂŒschendorf, B. Schweizer, and M. D. Taylor, vol. 28, pp. 65-75 (1996) |
1559:
to exactly one (distinct) column. These edges define a permutation matrix whose non-zero cells correspond to non-zero cells in
1912:
1671:
2759:
2525:
787:
105:
813:
states that any matrix with strictly positive entries can be made doubly stochastic by pre- and post-multiplication by
1347:{\displaystyle X-\lambda P={\frac {1}{12}}{\begin{pmatrix}7&0&3\\0&6&4\\3&4&3\end{pmatrix}}}
1857:
Falikman, D. I. (1981), "Proof of the van der
Waerden conjecture on the permanent of a doubly stochastic matrix",
2560:
1734:
877:
2107:
1600: ; to apply Birkhoff's theorem to the resulting square matrix; and at the end to additively recombine the
1219:
2324:
1961:
1123:{\displaystyle X={\frac {1}{12}}{\begin{pmatrix}7&0&5\\2&6&4\\3&6&3\end{pmatrix}}}
2399:
643:
2555:
2077:
1904:
1689:
1552:
1019:
will be a scalar multiple of a doubly stochastic matrix and will have at least one more zero cell than
772:
1757:
Gyires, B. (1980), "The common source of several inequalities concerning doubly stochastic matrices",
2659:
2530:
2444:
2764:
2362:
2042:
1663:
2799:
2728:
2610:
2470:
2067:
1954:
1820:
1555:
are satisfied, and that we can therefore find a set of edges in the graph which join each row in
881:
56:
2669:
2252:
2057:
851:
471:
368:
independent linear constraints specifying that the row and column sums all equal 1. (There are
272:
208:
897:
2615:
2352:
2202:
2197:
2032:
2007:
2002:
810:
266:
1655:
2809:
2167:
1997:
1977:
1870:
1843:
1812:
1793:
1770:
1629:
1209:{\displaystyle P={\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}}
847:
504:
440:
311:
241:
1941:
1922:
371:
342:
8:
2830:
2804:
2382:
2187:
2177:
1724:
W. B. Jurkat and H. J. Ryser, "Term Ranks and
Permanents of Nonnegative Matrices" (1967).
1656:
935:
823:
979:
be a doubly stochastic matrix. Then we will show that there exists a permutation matrix
400:
2881:
2835:
2825:
2779:
2774:
2703:
2639:
2505:
2242:
2237:
2172:
2162:
2027:
1897:
1634:
857:
531:
494:
1571:
There is a simple generalisation to matrices with more columns and rows such that the
2913:
2892:
2876:
2679:
2674:
2664:
2644:
2605:
2600:
2429:
2424:
2409:
2404:
2395:
2390:
2337:
2232:
2182:
2127:
2097:
2092:
2072:
2062:
2022:
1908:
1834:
1818:
Egorychev, G. P. (1981), "The solution of van der
Waerden's problem for permanents",
1667:
1624:
498:
200:
181:
2887:
2855:
2784:
2723:
2718:
2698:
2634:
2540:
2510:
2495:
2475:
2414:
2367:
2342:
2332:
2303:
2222:
2217:
2192:
2122:
2102:
2012:
1992:
1936:
1918:
1829:
1799:
Egorychev, G. P. (1981), "Proof of the van der
Waerden conjecture for permanents",
1788:(in Russian), Krasnoyarsk: Akad. Nauk SSSR Sibirsk. Otdel. Inst. Fiz., p. 12,
2480:
786:, and may not be unique. It is often described as a real-valued generalization of
633:{\displaystyle \theta _{1},\ldots ,\theta _{k}\geq 0,\sum _{i=1}^{k}\theta _{i}=1}
2585:
2520:
2500:
2485:
2465:
2449:
2347:
2278:
2268:
2227:
2112:
2082:
1866:
1839:
1808:
1789:
1766:
1363:
961:
814:
336:
232:
790:, where the correspondence is established through adjacency matrices of graphs.
2845:
2789:
2769:
2754:
2713:
2590:
2550:
2515:
2439:
2378:
2357:
2298:
2288:
2273:
2207:
2152:
2142:
2137:
2047:
1759:
Publicationes
Mathematicae Institutum Mathematicum Universitatis Debreceniensis
1903:. Encyclopedia of Mathematics and Its Applications. Vol. 108. Cambridge:
2907:
2850:
2708:
2649:
2580:
2570:
2565:
2490:
2419:
2293:
2283:
2212:
2132:
2117:
2052:
188:
51:
39:
2733:
2690:
2595:
2308:
2247:
2157:
2037:
804:
1713:
2575:
2545:
2313:
2147:
2017:
1023:. Accordingly we may successively reduce the number of non-zero cells in
465:
96:
35:
31:
1883:
2626:
2087:
1370:
are listed in one part and the columns in the other, and in which row
2860:
2434:
807:
is uniform if and only if its transition matrix is doubly stochastic.
771:
is known as a 'convex combination'.) A proof of the theorem based on
1704:
R. M. Caron, Xin Li, P. MikusiĆski, H. Sherwood, and M. D. Taylor,
2794:
187:
Indeed, any matrix that is both left and right stochastic must be
1946:
1529: ; and this is ≥ the corresponding sum in which the
803:
The stationary distribution of an irreducible aperiodic finite
997: ≠ 0. Thus if we let λ be the smallest
970:
932:, achieved by the matrix for which all entries are equal to
757:{\displaystyle X=\theta _{1}P_{1}+\cdots +\theta _{k}P_{k}.}
528:
are precisely the permutation matrices. In other words, if
1886:, Mathematical Optimization Society, retrieved 2012-08-19.
1658:
1593:
separate rows, each equal to the original row divided by
1942:
PlanetMath page on proof of
Birkhoffâvon Neumann theorem
1286:
1148:
1062:
1710:
Distributions with Fixed
Marginals and Related Topics
1255:
1222:
1136:
1040:
938:
900:
860:
826:
695:
646:
554:
534:
507:
474:
443:
403:
374:
345:
314:
275:
244:
211:
108:
59:
1896:
1786:Reshenie problemy van-der-Vardena dlya permanentov
1346:
1241:
1208:
1122:
952:
924:
866:
838:
756:
678:
632:
540:
520:
486:
456:
412:
389:
360:
327:
300:
257:
223:
170:{\displaystyle \sum _{i}x_{ij}=\sum _{j}x_{ij}=1,}
169:
99:, each of whose rows and columns sums to 1, i.e.,
87:
2905:
548:is a doubly stochastic matrix, then there exist
1937:PlanetMath page on Birkhoffâvon Neumann theorem
1733:
424:
1653:
180:Thus, a doubly stochastic matrix is both left
1962:
1400:in the graph. We want to express the sizes |
2536:Fundamental (linear differential equation)
1969:
1955:
1833:
1817:
1798:
971:Proof of the Birkhoffâvon Neumann theorem
1856:
1783:
1685:
1683:
1396:as the set of columns joined to rows in
1242:{\displaystyle \lambda ={\frac {2}{12}}}
2841:Matrix representation of conic sections
1894:
1700:
1698:
14:
2906:
1756:
1706:Nonsquare âdoubly stochasticâ matrices
1950:
1718:
1680:
1452: ≠ 0 are included in
1695:
782:This representation is known as the
194:
793:
679:{\displaystyle P_{1},\ldots ,P_{k}}
24:
1976:
1566:
1551:It follows that the conditions of
1408:| of the two sets in terms of the
784:Birkhoffâvon Neumann decomposition
776:
25:
2925:
1930:
2875:
846:, all bistochastic matrices are
308:-dimensional affine subspace of
231:doubly stochastic matrices is a
2743:Used in science and engineering
1389:be any set of rows, and define
1986:Explicitly constrained entries
1877:
1850:
1777:
1750:
1727:
1647:
1586:row of the original matrix by
990: ≠ 0 whenever
894:doubly stochastic matrices is
289:
276:
265:. Using the matrix entries as
82:
66:
13:
1:
2760:Fundamental (computer vision)
1640:
1463:is doubly stochastic; hence |
1899:Combinatorial matrix classes
1895:Brualdi, Richard A. (2006).
1835:10.1016/0001-8708(81)90044-X
1739:Jber. Deutsch. Math.-Verein.
1004:corresponding to a non-zero
878:Van der Waerden's conjecture
431:Birkhoffâvon Neumann theorem
425:Birkhoffâvon Neumann theorem
18:Birkhoffâvon Neumann theorem
7:
2526:Duplication and elimination
2325:eigenvalues or eigenvectors
1714:DOI:10.1214/lnms/1215452610
1618:
497:, and furthermore that the
437:) states that the polytope
10:
2930:
2459:With specific applications
2088:Discrete Fourier Transform
1905:Cambridge University Press
1385: ≠ 0. Let
198:
88:{\displaystyle X=(x_{ij})}
2869:
2818:
2750:CabibboâKobayashiâMaskawa
2742:
2688:
2624:
2458:
2377:Satisfying conditions on
2376:
2322:
2261:
1985:
1859:Akademiya Nauk Soyuza SSR
767:(Such a decomposition of
640:and permutation matrices
487:{\displaystyle n\times n}
301:{\displaystyle (n-1)^{2}}
224:{\displaystyle n\times n}
1784:EgoryÄev, G. P. (1980),
1692:, notes by GĂĄbor Hetyei.
1441:is 1, since all columns
925:{\displaystyle n!/n^{n}}
397:constraints rather than
44:doubly stochastic matrix
2108:Generalized permutation
1821:Advances in Mathematics
1654:Marshal, Olkin (1979).
1553:Hall's marriage theorem
1533:are limited to rows in
1374:is connected to column
773:Hall's marriage theorem
433:(often known simply as
2882:Mathematics portal
1737:(1926), "Aufgabe 45",
1735:van der Waerden, B. L.
1503:| is the sum over all
1467:| is the sum over all
1348:
1243:
1210:
1124:
954:
926:
868:
840:
758:
680:
634:
613:
542:
522:
488:
458:
414:
391:
362:
329:
302:
259:
225:
184:and right stochastic.
171:
89:
1544:| ≥ |
1366:in which the rows of
1349:
1244:
1211:
1125:
955:
927:
874:this is not the case.
869:
841:
759:
681:
635:
593:
543:
523:
521:{\displaystyle B_{n}}
489:
459:
457:{\displaystyle B_{n}}
415:
392:
363:
330:
328:{\displaystyle n^{2}}
303:
267:Cartesian coordinates
260:
258:{\displaystyle B_{n}}
226:
172:
90:
1630:Unistochastic matrix
1575:row sum is equal to
1253:
1220:
1134:
1038:
936:
898:
858:
824:
693:
644:
552:
532:
505:
495:permutation matrices
472:
441:
401:
390:{\displaystyle 2n-1}
372:
361:{\displaystyle 2n-1}
343:
312:
273:
242:
209:
106:
57:
2831:Linear independence
2078:Diagonally dominant
1865:(6): 931â938, 957,
1801:Akademiya Nauk SSSR
1607:rows into a single
1507:(whether or not in
1015: â λ
953:{\displaystyle 1/n}
839:{\displaystyle n=2}
48:bistochastic matrix
2836:Matrix exponential
2826:Jordan normal form
2660:Fisher information
2531:Euclidean distance
2445:Totally unimodular
1690:Birkhoff's theorem
1635:Birkhoff algorithm
1344:
1338:
1239:
1206:
1200:
1120:
1114:
950:
922:
864:
836:
811:Sinkhorn's theorem
754:
676:
630:
538:
518:
484:
454:
435:Birkhoff's theorem
413:{\displaystyle 2n}
410:
387:
358:
325:
298:
255:
221:
167:
144:
118:
85:
2901:
2900:
2893:Category:Matrices
2765:Fuzzy associative
2655:Doubly stochastic
2363:Positive-definite
2043:Block tridiagonal
1914:978-0-521-86565-4
1807:(6): 65â71, 225,
1673:978-0-12-473750-1
1625:Stochastic matrix
1279:
1237:
1055:
1011:, the difference
880:that the minimum
867:{\displaystyle n}
854:, but for larger
815:diagonal matrices
541:{\displaystyle X}
237:Birkhoff polytope
201:Birkhoff polytope
195:Birkhoff polytope
135:
109:
16:(Redirected from
2921:
2888:List of matrices
2880:
2879:
2856:Row echelon form
2800:State transition
2729:Seidel adjacency
2611:Totally positive
2471:Alternating sign
2068:Complex Hadamard
1971:
1964:
1957:
1948:
1947:
1926:
1902:
1887:
1881:
1875:
1873:
1854:
1848:
1846:
1837:
1815:
1796:
1781:
1775:
1773:
1765:(3â4): 291â304,
1754:
1748:
1746:
1731:
1725:
1722:
1716:
1702:
1693:
1687:
1678:
1677:
1661:
1651:
1542:
1520:
1501:
1484:
1457:
1394:
1353:
1351:
1350:
1345:
1343:
1342:
1280:
1272:
1248:
1246:
1245:
1240:
1238:
1230:
1215:
1213:
1212:
1207:
1205:
1204:
1129:
1127:
1126:
1121:
1119:
1118:
1056:
1048:
1034:For instance if
959:
957:
956:
951:
946:
931:
929:
928:
923:
921:
920:
911:
893:
873:
871:
870:
865:
845:
843:
842:
837:
794:Other properties
763:
761:
760:
755:
750:
749:
740:
739:
721:
720:
711:
710:
685:
683:
682:
677:
675:
674:
656:
655:
639:
637:
636:
631:
623:
622:
612:
607:
583:
582:
564:
563:
547:
545:
544:
539:
527:
525:
524:
519:
517:
516:
493:
491:
490:
485:
463:
461:
460:
455:
453:
452:
419:
417:
416:
411:
396:
394:
393:
388:
367:
365:
364:
359:
334:
332:
331:
326:
324:
323:
307:
305:
304:
299:
297:
296:
269:, it lies in an
264:
262:
261:
256:
254:
253:
230:
228:
227:
222:
176:
174:
173:
168:
157:
156:
143:
131:
130:
117:
94:
92:
91:
86:
81:
80:
34:, especially in
21:
2929:
2928:
2924:
2923:
2922:
2920:
2919:
2918:
2904:
2903:
2902:
2897:
2874:
2865:
2814:
2738:
2684:
2620:
2454:
2372:
2318:
2257:
2058:Centrosymmetric
1981:
1975:
1933:
1915:
1891:
1890:
1884:Fulkerson Prize
1882:
1878:
1855:
1851:
1782:
1778:
1755:
1751:
1732:
1728:
1723:
1719:
1703:
1696:
1688:
1681:
1674:
1652:
1648:
1643:
1621:
1605:
1598:
1591:
1580:
1569:
1567:Generalisations
1540:
1527:
1518:
1499:
1491:
1482:
1455:
1450:
1439:
1426:, the sum over
1413:
1392:
1383:
1364:bipartite graph
1337:
1336:
1331:
1326:
1320:
1319:
1314:
1309:
1303:
1302:
1297:
1292:
1282:
1281:
1271:
1254:
1251:
1250:
1229:
1221:
1218:
1217:
1199:
1198:
1193:
1188:
1182:
1181:
1176:
1171:
1165:
1164:
1159:
1154:
1144:
1143:
1135:
1132:
1131:
1113:
1112:
1107:
1102:
1096:
1095:
1090:
1085:
1079:
1078:
1073:
1068:
1058:
1057:
1047:
1039:
1036:
1035:
1009:
1002:
995:
988:
973:
962:Fulkerson Prize
942:
937:
934:
933:
916:
912:
907:
899:
896:
895:
885:
859:
856:
855:
852:orthostochastic
825:
822:
821:
796:
788:KĆnig's theorem
745:
741:
735:
731:
716:
712:
706:
702:
694:
691:
690:
670:
666:
651:
647:
645:
642:
641:
618:
614:
608:
597:
578:
574:
559:
555:
553:
550:
549:
533:
530:
529:
512:
508:
506:
503:
502:
473:
470:
469:
448:
444:
442:
439:
438:
427:
402:
399:
398:
373:
370:
369:
344:
341:
340:
337:Euclidean space
319:
315:
313:
310:
309:
292:
288:
274:
271:
270:
249:
245:
243:
240:
239:
233:convex polytope
210:
207:
206:
203:
197:
149:
145:
139:
123:
119:
113:
107:
104:
103:
95:of nonnegative
73:
69:
58:
55:
54:
28:
27:A square matrix
23:
22:
15:
12:
11:
5:
2927:
2917:
2916:
2899:
2898:
2896:
2895:
2890:
2885:
2870:
2867:
2866:
2864:
2863:
2858:
2853:
2848:
2846:Perfect matrix
2843:
2838:
2833:
2828:
2822:
2820:
2816:
2815:
2813:
2812:
2807:
2802:
2797:
2792:
2787:
2782:
2777:
2772:
2767:
2762:
2757:
2752:
2746:
2744:
2740:
2739:
2737:
2736:
2731:
2726:
2721:
2716:
2711:
2706:
2701:
2695:
2693:
2686:
2685:
2683:
2682:
2677:
2672:
2667:
2662:
2657:
2652:
2647:
2642:
2637:
2631:
2629:
2622:
2621:
2619:
2618:
2616:Transformation
2613:
2608:
2603:
2598:
2593:
2588:
2583:
2578:
2573:
2568:
2563:
2558:
2553:
2548:
2543:
2538:
2533:
2528:
2523:
2518:
2513:
2508:
2503:
2498:
2493:
2488:
2483:
2478:
2473:
2468:
2462:
2460:
2456:
2455:
2453:
2452:
2447:
2442:
2437:
2432:
2427:
2422:
2417:
2412:
2407:
2402:
2393:
2387:
2385:
2374:
2373:
2371:
2370:
2365:
2360:
2355:
2353:Diagonalizable
2350:
2345:
2340:
2335:
2329:
2327:
2323:Conditions on
2320:
2319:
2317:
2316:
2311:
2306:
2301:
2296:
2291:
2286:
2281:
2276:
2271:
2265:
2263:
2259:
2258:
2256:
2255:
2250:
2245:
2240:
2235:
2230:
2225:
2220:
2215:
2210:
2205:
2203:Skew-symmetric
2200:
2198:Skew-Hermitian
2195:
2190:
2185:
2180:
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2045:
2040:
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2033:Block-diagonal
2030:
2025:
2020:
2015:
2010:
2008:Anti-symmetric
2005:
2003:Anti-Hermitian
2000:
1995:
1989:
1987:
1983:
1982:
1974:
1973:
1966:
1959:
1951:
1945:
1944:
1939:
1932:
1931:External links
1929:
1928:
1927:
1913:
1889:
1888:
1876:
1861:(in Russian),
1849:
1828:(3): 299â305,
1803:(in Russian),
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199:Main article:
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2851:Pseudoinverse
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2821:
2819:Related terms
2817:
2811:
2810:Z (chemistry)
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2168:Pentadiagonal
2166:
2164:
2161:
2159:
2156:
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2149:
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2019:
2016:
2014:
2011:
2009:
2006:
2004:
2001:
1999:
1998:Anti-diagonal
1996:
1994:
1991:
1990:
1988:
1984:
1979:
1972:
1967:
1965:
1960:
1958:
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1952:
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1916:
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1479: â
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917:
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908:
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888:
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879:
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861:
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849:
848:unistochastic
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619:
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594:
590:
587:
584:
579:
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571:
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556:
535:
513:
509:
500:
496:
481:
478:
475:
467:
449:
445:
436:
432:
422:
407:
404:
384:
381:
378:
375:
355:
352:
349:
346:
338:
335:-dimensional
320:
316:
293:
285:
282:
279:
268:
250:
246:
238:
235:known as the
234:
218:
215:
212:
205:The class of
202:
192:
190:
185:
183:
164:
161:
158:
153:
150:
146:
140:
136:
132:
127:
124:
120:
114:
110:
102:
101:
100:
98:
77:
74:
70:
63:
60:
53:
52:square matrix
49:
46:(also called
45:
41:
40:combinatorics
37:
33:
19:
2873:
2805:Substitution
2691:graph theory
2654:
2188:Quaternionic
2178:Persymmetric
1898:
1879:
1862:
1858:
1852:
1825:
1819:
1804:
1800:
1785:
1779:
1762:
1758:
1752:
1742:
1738:
1729:
1720:
1709:
1705:
1657:
1649:
1613:
1608:
1601:
1594:
1587:
1583:
1576:
1572:
1570:
1560:
1556:
1550:
1545:
1538:
1534:
1530:
1523:
1516:
1512:
1508:
1504:
1497:
1495:
1487:
1480:
1476:
1472:
1468:
1464:
1460:
1453:
1446:
1442:
1435:
1431:
1427:
1423:
1419:
1417:
1409:
1405:
1401:
1397:
1390:
1386:
1379:
1375:
1371:
1367:
1362:Construct a
1358:
1357:
1356:
1033:
1028:
1024:
1020:
1016:
1012:
1005:
998:
991:
984:
980:
976:
974:
967:
890:
886:
805:Markov chain
783:
781:
768:
766:
434:
430:
428:
236:
204:
186:
179:
97:real numbers
47:
43:
29:
2780:Hamiltonian
2704:Biadjacency
2640:Correlation
2556:Householder
2506:Commutation
2243:Vandermonde
2238:Tridiagonal
2173:Permutation
2163:Nonnegative
2148:Matrix unit
2028:Bisymmetric
1662:. pp.
1496:Meanwhile |
466:convex hull
339:defined by
36:probability
32:mathematics
2680:Transition
2675:Stochastic
2645:Covariance
2627:statistics
2606:Symplectic
2601:Similarity
2430:Unimodular
2425:Orthogonal
2410:Involutory
2405:Invertible
2400:Projection
2396:Idempotent
2338:Convergent
2233:Triangular
2183:Polynomial
2128:Hessenberg
2098:Equivalent
2093:Elementary
2073:Copositive
2063:Conference
2023:Bidiagonal
1923:1106.05001
1641:References
1511:) and all
1445:for which
1418:For every
983:such that
884:among all
686:such that
182:stochastic
2861:Wronskian
2785:Irregular
2775:Gell-Mann
2724:Laplacian
2719:Incidence
2699:Adjacency
2670:Precision
2635:Centering
2541:Generator
2511:Confusion
2496:Circulant
2476:Augmented
2435:Unipotent
2415:Nilpotent
2391:Congruent
2368:Stieltjes
2343:Defective
2333:Companion
2304:Redheffer
2223:Symmetric
2218:Sylvester
2193:Signature
2123:Hermitian
2103:Frobenius
2013:Arrowhead
1993:Alternant
1537:. Hence |
1263:λ
1260:−
1224:λ
882:permanent
800:entries).
775:is given
733:θ
726:⋯
704:θ
661:…
616:θ
595:∑
585:≥
576:θ
569:…
557:θ
479:×
382:−
353:−
283:−
216:×
137:∑
111:∑
2914:Matrices
2908:Category
2689:Used in
2625:Used in
2586:Rotation
2561:Jacobian
2521:Distance
2501:Cofactor
2486:Carleman
2466:Adjugate
2450:Weighing
2383:inverses
2379:products
2348:Definite
2279:Identity
2269:Exchange
2262:Constant
2228:Toeplitz
2113:Hadamard
2083:Diagonal
1619:See also
964:in 1982.
499:vertices
2790:Overlap
2755:Density
2714:Edmonds
2591:Seifert
2551:Hessian
2516:Coxeter
2440:Unitary
2358:Hurwitz
2289:Of ones
2274:Hilbert
2208:Skyline
2153:Metzler
2143:Logical
2138:Integer
2048:Boolean
1980:classes
1871:0625097
1844:0642395
1813:0638007
1794:0602332
1771:0604006
1404:| and |
1249:, and
464:is the
50:) is a
2709:Degree
2650:Design
2581:Random
2571:Payoff
2566:Moment
2491:Cartan
2481:BĂ©zout
2420:Normal
2294:Pascal
2284:Lehmer
2213:Sparse
2133:Hollow
2118:Hankel
2053:Cauchy
1978:Matrix
1921:
1911:
1869:
1842:
1811:
1792:
1769:
1708:, in:
1670:
1459:, and
1359:Proof:
1130:then
189:square
2770:Gamma
2734:Tutte
2596:Shear
2309:Shift
2299:Pauli
2248:Walsh
2158:Moore
2038:Block
1745:: 117
1611:row.
1541:'
1519:'
1500:'
1483:'
1456:'
1393:'
777:below
2576:Pick
2546:Gram
2314:Zero
2018:Band
1909:ISBN
1668:ISBN
1378:iff
975:Let
850:and
820:For
501:of
429:The
42:, a
38:and
2665:Hat
2398:or
2381:or
1919:Zbl
1830:doi
1563:.
1548:|.
1522:of
1515:in
1486:of
1434:of
1430:in
1422:in
30:In
2910::
1917:.
1907:.
1867:MR
1863:29
1840:MR
1838:,
1826:42
1824:,
1816:.
1809:MR
1805:22
1797:.
1790:MR
1767:MR
1763:27
1761:,
1743:35
1741:,
1697:^
1682:^
1666:.
1526:ij
1493:.
1490:ij
1475:,
1449:ij
1438:ij
1432:A'
1415:.
1412:ij
1406:A'
1382:ij
1354:.
1277:12
1235:12
1216:,
1053:12
1031:.
1008:ij
1001:ij
994:ij
987:ij
889:Ă
779:.
2795:S
2253:Z
1970:e
1963:t
1956:v
1925:.
1874:.
1847:.
1832::
1774:.
1747:.
1676:.
1664:8
1609:i
1604:i
1602:r
1597:i
1595:r
1590:i
1588:r
1584:i
1579:i
1577:r
1573:i
1561:X
1557:X
1546:A
1539:A
1535:A
1531:i
1524:x
1517:A
1513:j
1509:A
1505:i
1498:A
1488:x
1481:A
1477:j
1473:A
1469:i
1465:A
1461:X
1454:A
1447:x
1443:j
1436:x
1428:j
1424:A
1420:i
1410:x
1402:A
1398:A
1391:A
1387:A
1380:x
1376:j
1372:i
1368:X
1340:)
1334:3
1329:4
1324:3
1317:4
1312:6
1307:0
1300:3
1295:0
1290:7
1284:(
1274:1
1269:=
1266:P
1257:X
1232:2
1227:=
1202:)
1196:0
1191:1
1186:0
1179:0
1174:0
1169:1
1162:1
1157:0
1152:0
1146:(
1141:=
1138:P
1116:)
1110:3
1105:6
1100:3
1093:4
1088:6
1083:2
1076:5
1071:0
1066:7
1060:(
1050:1
1045:=
1042:X
1029:X
1025:X
1021:X
1017:P
1013:X
1006:p
999:x
992:p
985:x
981:P
977:X
948:n
944:/
940:1
918:n
914:n
909:/
905:!
902:n
891:n
887:n
862:n
834:2
831:=
828:n
817:.
769:X
752:.
747:k
743:P
737:k
729:+
723:+
718:1
714:P
708:1
700:=
697:X
672:k
668:P
664:,
658:,
653:1
649:P
628:1
625:=
620:i
610:k
605:1
602:=
599:i
591:,
588:0
580:k
572:,
566:,
561:1
536:X
514:n
510:B
482:n
476:n
450:n
446:B
408:n
405:2
385:1
379:n
376:2
356:1
350:n
347:2
321:2
317:n
294:2
290:)
286:1
280:n
277:(
251:n
247:B
219:n
213:n
165:,
162:1
159:=
154:j
151:i
147:x
141:j
133:=
128:j
125:i
121:x
115:i
83:)
78:j
75:i
71:x
67:(
64:=
61:X
20:)
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