934:
2897:
606:
1471:
929:{\displaystyle {\begin{bmatrix}B_{11}&B_{12}&0&\cdots &\cdots &0\\B_{21}&B_{22}&B_{23}&\ddots &\ddots &\vdots \\0&B_{32}&B_{33}&B_{34}&\ddots &\vdots \\\vdots &\ddots &B_{43}&B_{44}&B_{45}&0\\\vdots &\ddots &\ddots &B_{54}&B_{55}&B_{56}\\0&\cdots &\cdots &0&B_{65}&B_{66}\end{bmatrix}}}
1197:
1186:
1716:
1466:{\displaystyle {\begin{bmatrix}A_{11}&A_{12}&A_{13}&0&\cdots &0\\&A_{22}&A_{23}&A_{24}&\ddots &\vdots \\&&A_{33}&A_{34}&A_{35}&0\\&&&A_{44}&A_{45}&A_{46}\\&sym&&&A_{55}&A_{56}\\&&&&&A_{66}\end{bmatrix}}.}
568:
problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the banded property corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided – for instance, banded
1737:
As sparse matrices lend themselves to more efficient computation than dense matrices, as well as in more efficient utilization of computer storage, there has been much research focused on finding ways to minimise the bandwidth (or directly minimise the fill-in) by applying permutations to the
1730:. A band matrix can be likened in complexity to a rectangular matrix whose row dimension is equal to the bandwidth of the band matrix. Thus the work involved in performing operations such as multiplication falls significantly, often leading to huge savings in terms of calculation time and
945:
1482:
242:
572:
Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the
1181:{\displaystyle {\begin{bmatrix}0&B_{11}&B_{12}\\B_{21}&B_{22}&B_{23}\\B_{32}&B_{33}&B_{34}\\B_{43}&B_{44}&B_{45}\\B_{54}&B_{55}&B_{56}\\B_{65}&B_{66}&0\end{bmatrix}}.}
1711:{\displaystyle {\begin{bmatrix}A_{11}&A_{12}&A_{13}\\A_{22}&A_{23}&A_{24}\\A_{33}&A_{34}&A_{35}\\A_{44}&A_{45}&A_{46}\\A_{55}&A_{56}&0\\A_{66}&0&0\end{bmatrix}}.}
112:
2555:
382:
340:
2769:
1988:
2860:
1940:
1917:
1874:
2779:
2545:
1750:
1191:
A further saving is possible when the matrix is symmetric. For example, consider a symmetric 6-by-6 matrix with an upper bandwidth of 2:
1855:
2580:
2127:
1726:
From a computational point of view, working with band matrices is always preferential to working with similarly dimensioned
237:{\displaystyle a_{i,j}=0\quad {\mbox{if}}\quad j<i-k_{1}\quad {\mbox{ or }}\quad j>i+k_{2};\quad k_{1},k_{2}\geq 0.\,}
1756:
The problem of finding a representation of a matrix with minimal bandwidth by means of permutations of rows and columns is
2344:
1981:
1742:
2419:
1961:
2575:
2097:
2679:
2550:
2464:
2933:
2784:
2674:
2382:
2062:
92:). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants
2819:
2748:
2630:
2490:
2087:
1974:
2689:
2272:
2077:
1731:
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2372:
2222:
2217:
2052:
2027:
2022:
516:
2896:
2829:
2187:
2017:
1997:
578:
35:
593:
Band matrices are usually stored by storing the diagonals in the band; the rest is implicitly zero.
2850:
2824:
2402:
2207:
2197:
1905:
577:
of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying
2901:
2855:
2845:
2799:
2794:
2723:
2659:
2525:
2262:
2257:
2192:
2182:
2047:
597:
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531:
429:
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2912:
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2625:
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2415:
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2147:
2117:
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2092:
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1936:
1913:
1870:
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565:
510:
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452:
2907:
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2012:
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2132:
2102:
1774:
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408:
17:
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561:
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2728:
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2585:
2510:
2439:
2313:
2303:
2232:
2152:
2137:
2072:
1901:
544:
540:, also called "variable band matrix" – a generalization of band matrix
55:
47:
2753:
2710:
2615:
2328:
2267:
2177:
2057:
1884:
Feige, Uriel (2000), "Coping with the NP-Hardness of the Graph
Bandwidth Problem",
525:
521:
1928:
585:) to such a matrix results in the band being filled in by many non-zero elements.
2595:
2565:
2333:
2167:
1956:
574:
31:
2646:
2107:
2880:
2454:
1893:
2814:
1888:, Lecture Notes in Computer Science, vol. 1851, pp. 129–145,
1757:
1966:
1738:
matrix, or other such equivalence or similarity transformations.
1970:
1927:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007),
547:
are constant tridiagonal matrices, and are thus band matrices.
1962:
A tutorial on banded matrices and other sparse matrix formats
569:
matrices exist where every element in the band is nonzero.
1935:(3rd ed.), New York: Cambridge University Press,
1753:
performs better. There are many other methods in use.
1491:
1206:
954:
615:
171:
140:
1485:
1200:
948:
609:
348:
309:
115:
27:
Matrix with non-zero elements only in a diagonal band
2838:
2762:
2708:
2644:
2478:
2396:
2342:
2281:
2005:
1957:Information pertaining to LAPACK and band matrices
1933:Numerical Recipes: The Art of Scientific Computing
1869:, Society for Industrial and Applied Mathematics,
1710:
1465:
1180:
928:
376:
334:
236:
50:whose non-zero entries are confined to a diagonal
475:−1, one obtains the definition of an upper
1745:can be used to reduce the bandwidth of a sparse
1749:. There are, however, matrices for which the
1982:
1793:
449:= 2 one has a pentadiagonal matrix and so on.
8:
1476:This matrix is stored as the 6-by-3 matrix:
58:and zero or more diagonals on either side.
2556:Fundamental (linear differential equation)
1989:
1975:
1967:
1912:(3rd ed.), Baltimore: Johns Hopkins,
499:= 0 one obtains a lower triangular matrix.
1681:
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1636:
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1867:Direct Methods for Sparse Linear Systems
1805:
2861:Matrix representation of conic sections
1786:
1848:An Introduction to Numerical Analysis
1829:
1817:
7:
600:has bandwidth 1. The 6-by-6 matrix
299:; in other words, it is the number
25:
2895:
285:of the matrix is the maximum of
2763:Used in science and engineering
1751:reverse Cuthill–McKee algorithm
939:is stored as the 6-by-3 matrix
203:
177:
169:
146:
138:
2006:Explicitly constrained entries
364:
350:
1:
2780:Fundamental (computer vision)
1846:Atkinson, Kendall E. (1989),
1886:Algorithm Theory - SWAT 2000
1722:Band form of sparse matrices
2546:Duplication and elimination
2345:eigenvalues or eigenvectors
2950:
2479:With specific applications
2108:Discrete Fourier Transform
1865:Davis, Timothy A. (2006),
513:when bandwidth is limited.
377:{\displaystyle |i-j|>k}
18:Bandwidth (linear algebra)
2889:
2770:Cabibbo–Kobayashi–Maskawa
2397:Satisfying conditions on
1850:, John Wiley & Sons,
1794:Golub & Van Loan 1996
335:{\displaystyle a_{i,j}=0}
2128:Generalized permutation
1894:10.1007/3-540-44985-X_2
1743:Cuthill–McKee algorithm
517:Block diagonal matrices
2902:Mathematics portal
1712:
1467:
1182:
930:
378:
336:
238:
71:Formally, consider an
1713:
1468:
1183:
931:
379:
337:
239:
1906:Van Loan, Charles F.
1483:
1198:
946:
607:
581:(or equivalently an
579:Gaussian elimination
346:
307:
277:, respectively. The
247:then the quantities
113:
2851:Linear independence
2098:Diagonally dominant
1910:Matrix Computations
506:Hessenberg matrices
453:Triangular matrices
432:, with bandwidth 1.
414:A band matrix with
411:, with bandwidth 0.
393:A band matrix with
2856:Matrix exponential
2846:Jordan normal form
2680:Fisher information
2551:Euclidean distance
2465:Totally unimodular
1708:
1699:
1463:
1454:
1178:
1169:
926:
920:
598:tridiagonal matrix
558:numerical analysis
532:Jordan normal form
430:tridiagonal matrix
374:
332:
234:
175:
144:
2921:
2920:
2913:Category:Matrices
2785:Fuzzy associative
2675:Doubly stochastic
2383:Positive-definite
2063:Block tridiagonal
1942:978-0-521-88068-8
1919:978-0-8018-5414-9
1876:978-0-898716-13-9
566:finite difference
511:Toeplitz matrices
477:triangular matrix
174:
143:
54:, comprising the
16:(Redirected from
2941:
2908:List of matrices
2900:
2899:
2876:Row echelon form
2820:State transition
2749:Seidel adjacency
2631:Totally positive
2491:Alternating sign
2088:Complex Hadamard
1991:
1984:
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1968:
1945:
1922:
1896:
1879:
1860:
1833:
1827:
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1803:
1797:
1791:
1747:symmetric matrix
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1321:
1311:
1310:
1296:
1295:
1284:
1283:
1272:
1271:
1261:
1242:
1241:
1230:
1229:
1218:
1217:
1187:
1185:
1184:
1179:
1174:
1173:
1161:
1160:
1149:
1148:
1135:
1134:
1123:
1122:
1111:
1110:
1097:
1096:
1085:
1084:
1073:
1072:
1059:
1058:
1047:
1046:
1035:
1034:
1021:
1020:
1009:
1008:
997:
996:
983:
982:
971:
970:
935:
933:
932:
927:
925:
924:
917:
916:
905:
904:
871:
870:
859:
858:
847:
846:
813:
812:
801:
800:
789:
788:
755:
754:
743:
742:
731:
730:
697:
696:
685:
684:
673:
672:
639:
638:
627:
626:
583:LU decomposition
560:, matrices from
543:The inverses of
504:Upper and lower
383:
381:
380:
375:
367:
353:
341:
339:
338:
333:
325:
324:
283:
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275:
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267:
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243:
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235:
226:
225:
213:
212:
199:
198:
176:
172:
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167:
145:
141:
131:
130:
21:
2949:
2948:
2944:
2943:
2942:
2940:
2939:
2938:
2934:Sparse matrices
2924:
2923:
2922:
2917:
2894:
2885:
2834:
2758:
2704:
2640:
2474:
2392:
2338:
2277:
2078:Centrosymmetric
2001:
1995:
1953:
1943:
1926:
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1900:
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1775:Graph bandwidth
1770:Diagonal matrix
1766:
1728:square matrices
1724:
1698:
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1407:
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1383:
1381:
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1369:
1359:
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1323:
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1308:
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1302:
1297:
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1259:
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1168:
1167:
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1114:
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1086:
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1038:
1036:
1026:
1023:
1022:
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1010:
1000:
998:
988:
985:
984:
974:
972:
962:
960:
950:
944:
943:
919:
918:
908:
906:
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894:
889:
884:
879:
873:
872:
862:
860:
850:
848:
838:
836:
831:
826:
820:
819:
814:
804:
802:
792:
790:
780:
778:
773:
767:
766:
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746:
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734:
732:
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708:
703:
698:
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674:
664:
661:
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655:
650:
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640:
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628:
618:
611:
605:
604:
596:For example, a
591:
554:
545:Lehmer matrices
498:
487:
481:similarly, for
470:
463:
448:
441:
427:
420:
409:diagonal matrix
406:
399:
390:
344:
343:
310:
305:
304:
298:
291:
280:
279:
273:upper bandwidth
272:
271:
265:lower bandwidth
264:
263:
261:are called the
260:
253:
217:
204:
190:
159:
116:
111:
110:
105:
98:
91:
69:
64:
34:, particularly
28:
23:
22:
15:
12:
11:
5:
2947:
2945:
2937:
2936:
2926:
2925:
2919:
2918:
2916:
2915:
2910:
2905:
2890:
2887:
2886:
2884:
2883:
2878:
2873:
2868:
2866:Perfect matrix
2863:
2858:
2853:
2848:
2842:
2840:
2836:
2835:
2833:
2832:
2827:
2822:
2817:
2812:
2807:
2802:
2797:
2792:
2787:
2782:
2777:
2772:
2766:
2764:
2760:
2759:
2757:
2756:
2751:
2746:
2741:
2736:
2731:
2726:
2721:
2715:
2713:
2706:
2705:
2703:
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2662:
2657:
2651:
2649:
2642:
2641:
2639:
2638:
2636:Transformation
2633:
2628:
2623:
2618:
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:
2482:
2480:
2476:
2475:
2473:
2472:
2467:
2462:
2457:
2452:
2447:
2442:
2437:
2432:
2427:
2422:
2413:
2407:
2405:
2394:
2393:
2391:
2390:
2385:
2380:
2375:
2373:Diagonalizable
2370:
2365:
2360:
2355:
2349:
2347:
2343:Conditions on
2340:
2339:
2337:
2336:
2331:
2326:
2321:
2316:
2311:
2306:
2301:
2296:
2291:
2285:
2283:
2279:
2278:
2276:
2275:
2270:
2265:
2260:
2255:
2250:
2245:
2240:
2235:
2230:
2225:
2223:Skew-symmetric
2220:
2218:Skew-Hermitian
2215:
2210:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2155:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2110:
2105:
2100:
2095:
2090:
2085:
2080:
2075:
2070:
2065:
2060:
2055:
2053:Block-diagonal
2050:
2045:
2040:
2035:
2030:
2028:Anti-symmetric
2025:
2023:Anti-Hermitian
2020:
2015:
2009:
2007:
2003:
2002:
1996:
1994:
1993:
1986:
1979:
1971:
1965:
1964:
1959:
1952:
1951:External links
1949:
1948:
1947:
1941:
1924:
1918:
1902:Golub, Gene H.
1898:
1881:
1875:
1862:
1856:
1841:
1838:
1835:
1834:
1822:
1810:
1808:, p. 527.
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1785:
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1374:
1370:
1366:
1362:
1358:
1354:
1351:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1312:
1309:
1306:
1303:
1301:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1260:
1257:
1254:
1252:
1249:
1247:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1207:
1205:
1189:
1188:
1177:
1172:
1166:
1163:
1159:
1155:
1151:
1147:
1143:
1139:
1138:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1101:
1100:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1062:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1025:
1024:
1019:
1015:
1011:
1007:
1003:
999:
995:
991:
987:
986:
981:
977:
973:
969:
965:
961:
959:
956:
955:
953:
937:
936:
923:
915:
911:
907:
903:
899:
895:
893:
890:
888:
885:
883:
880:
878:
875:
874:
869:
865:
861:
857:
853:
849:
845:
841:
837:
835:
832:
830:
827:
825:
822:
821:
818:
815:
811:
807:
803:
799:
795:
791:
787:
783:
779:
777:
774:
772:
769:
768:
765:
762:
760:
757:
753:
749:
745:
741:
737:
733:
729:
725:
721:
719:
716:
715:
712:
709:
707:
704:
702:
699:
695:
691:
687:
683:
679:
675:
671:
667:
663:
662:
659:
656:
654:
651:
649:
646:
644:
641:
637:
633:
629:
625:
621:
617:
616:
614:
590:
587:
562:finite element
553:
550:
549:
548:
541:
538:skyline matrix
534:
528:
526:shear matrices
522:Shift matrices
519:
514:
508:
502:
501:
500:
496:
485:
479:
468:
461:
450:
446:
439:
433:
425:
418:
412:
404:
397:
389:
386:
373:
370:
366:
362:
359:
356:
352:
331:
328:
323:
320:
317:
313:
296:
289:
258:
251:
245:
244:
232:
229:
224:
220:
216:
211:
207:
202:
197:
193:
189:
186:
183:
180:
173: or
166:
162:
158:
155:
152:
149:
137:
134:
129:
126:
123:
119:
103:
96:
87:
68:
65:
63:
60:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2946:
2935:
2932:
2931:
2929:
2914:
2911:
2909:
2906:
2904:
2903:
2898:
2892:
2891:
2888:
2882:
2879:
2877:
2874:
2872:
2871:Pseudoinverse
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2847:
2844:
2843:
2841:
2839:Related terms
2837:
2831:
2830:Z (chemistry)
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2767:
2765:
2761:
2755:
2752:
2750:
2747:
2745:
2742:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2716:
2714:
2712:
2707:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2652:
2650:
2648:
2643:
2637:
2634:
2632:
2629:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2483:
2481:
2477:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2421:
2417:
2414:
2412:
2409:
2408:
2406:
2404:
2400:
2395:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2350:
2348:
2346:
2341:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2290:
2287:
2286:
2284:
2280:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2188:Pentadiagonal
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2111:
2109:
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2084:
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2064:
2061:
2059:
2056:
2054:
2051:
2049:
2046:
2044:
2041:
2039:
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2019:
2018:Anti-diagonal
2016:
2014:
2011:
2010:
2008:
2004:
1999:
1992:
1987:
1985:
1980:
1978:
1973:
1972:
1969:
1963:
1960:
1958:
1955:
1954:
1950:
1944:
1938:
1934:
1930:
1929:"Section 2.4"
1925:
1921:
1915:
1911:
1907:
1903:
1899:
1895:
1891:
1887:
1882:
1878:
1872:
1868:
1863:
1859:
1857:0-471-62489-6
1853:
1849:
1844:
1843:
1839:
1831:
1826:
1823:
1819:
1814:
1811:
1807:
1806:Atkinson 1989
1802:
1799:
1795:
1790:
1787:
1780:
1776:
1773:
1771:
1768:
1767:
1763:
1761:
1759:
1754:
1752:
1748:
1744:
1739:
1735:
1733:
1729:
1721:
1705:
1700:
1694:
1689:
1682:
1678:
1670:
1663:
1659:
1651:
1647:
1637:
1633:
1625:
1621:
1613:
1609:
1599:
1595:
1587:
1583:
1575:
1571:
1561:
1557:
1549:
1545:
1537:
1533:
1523:
1519:
1511:
1507:
1499:
1495:
1488:
1479:
1478:
1477:
1460:
1455:
1447:
1443:
1428:
1424:
1416:
1412:
1404:
1401:
1398:
1388:
1384:
1376:
1372:
1364:
1360:
1349:
1342:
1338:
1330:
1326:
1318:
1314:
1304:
1299:
1292:
1288:
1280:
1276:
1268:
1264:
1255:
1250:
1245:
1238:
1234:
1226:
1222:
1214:
1210:
1203:
1194:
1193:
1192:
1175:
1170:
1164:
1157:
1153:
1145:
1141:
1131:
1127:
1119:
1115:
1107:
1103:
1093:
1089:
1081:
1077:
1069:
1065:
1055:
1051:
1043:
1039:
1031:
1027:
1017:
1013:
1005:
1001:
993:
989:
979:
975:
967:
963:
957:
951:
942:
941:
940:
921:
913:
909:
901:
897:
891:
886:
881:
876:
867:
863:
855:
851:
843:
839:
833:
828:
823:
816:
809:
805:
797:
793:
785:
781:
775:
770:
763:
758:
751:
747:
739:
735:
727:
723:
717:
710:
705:
700:
693:
689:
681:
677:
669:
665:
657:
652:
647:
642:
635:
631:
623:
619:
612:
603:
602:
601:
599:
594:
588:
586:
584:
580:
576:
570:
567:
563:
559:
551:
546:
542:
539:
535:
533:
529:
527:
523:
520:
518:
515:
512:
509:
507:
503:
495:
491:
484:
480:
478:
474:
467:
460:
456:
455:
454:
451:
445:
438:
434:
431:
424:
417:
413:
410:
403:
396:
392:
391:
387:
385:
371:
368:
360:
357:
354:
329:
326:
321:
318:
315:
311:
302:
295:
288:
284:
276:
268:
257:
250:
230:
227:
222:
218:
214:
209:
205:
200:
195:
191:
187:
184:
181:
178:
164:
160:
156:
153:
150:
147:
135:
132:
127:
124:
121:
117:
109:
108:
107:
102:
95:
90:
86:
82:
78:
74:
66:
61:
59:
57:
56:main diagonal
53:
49:
48:sparse matrix
45:
44:banded matrix
41:
37:
36:matrix theory
33:
19:
2893:
2825:Substitution
2711:graph theory
2208:Quaternionic
2198:Persymmetric
2037:
1932:
1909:
1885:
1866:
1847:
1825:
1813:
1801:
1789:
1755:
1740:
1736:
1725:
1475:
1190:
938:
595:
592:
589:Band storage
571:
555:
552:Applications
530:Matrices in
493:
489:
482:
472:
465:
458:
443:
436:
422:
415:
401:
394:
300:
293:
286:
278:
270:
262:
255:
248:
246:
100:
93:
88:
84:
80:
76:
72:
70:
51:
43:
39:
29:
2800:Hamiltonian
2724:Biadjacency
2660:Correlation
2576:Householder
2526:Commutation
2263:Vandermonde
2258:Tridiagonal
2193:Permutation
2183:Nonnegative
2168:Matrix unit
2048:Bisymmetric
575:square root
62:Band matrix
40:band matrix
32:mathematics
2700:Transition
2695:Stochastic
2665:Covariance
2647:statistics
2626:Symplectic
2621:Similarity
2450:Unimodular
2445:Orthogonal
2430:Involutory
2425:Invertible
2420:Projection
2416:Idempotent
2358:Convergent
2253:Triangular
2203:Polynomial
2148:Hessenberg
2118:Equivalent
2113:Elementary
2093:Copositive
2083:Conference
2043:Bidiagonal
1840:References
1830:Feige 2000
1818:Davis 2006
1732:complexity
492:−1,
303:such that
2881:Wronskian
2805:Irregular
2795:Gell-Mann
2744:Laplacian
2739:Incidence
2719:Adjacency
2690:Precision
2655:Centering
2561:Generator
2531:Confusion
2516:Circulant
2496:Augmented
2455:Unipotent
2435:Nilpotent
2411:Congruent
2388:Stieltjes
2363:Defective
2353:Companion
2324:Redheffer
2243:Symmetric
2238:Sylvester
2213:Signature
2143:Hermitian
2123:Frobenius
2033:Arrowhead
2013:Alternant
1796:, §1.2.1.
1305:⋮
1300:⋱
1251:⋯
887:⋯
882:⋯
834:⋱
829:⋱
824:⋮
776:⋱
771:⋮
764:⋮
759:⋱
711:⋮
706:⋱
701:⋱
653:⋯
648:⋯
428:= 1 is a
407:= 0 is a
358:−
281:bandwidth
228:≥
157:−
67:Bandwidth
2928:Category
2709:Used in
2645:Used in
2606:Rotation
2581:Jacobian
2541:Distance
2521:Cofactor
2506:Carleman
2486:Adjugate
2470:Weighing
2403:inverses
2399:products
2368:Definite
2299:Identity
2289:Exchange
2282:Constant
2248:Toeplitz
2133:Hadamard
2103:Diagonal
1908:(1996),
1764:See also
388:Examples
2810:Overlap
2775:Density
2734:Edmonds
2611:Seifert
2571:Hessian
2536:Coxeter
2460:Unitary
2378:Hurwitz
2309:Of ones
2294:Hilbert
2228:Skyline
2173:Metzler
2163:Logical
2158:Integer
2068:Boolean
2000:classes
1820:, §7.7.
1758:NP-hard
79:matrix
2729:Degree
2670:Design
2601:Random
2591:Payoff
2586:Moment
2511:Cartan
2501:BĂ©zout
2440:Normal
2314:Pascal
2304:Lehmer
2233:Sparse
2153:Hollow
2138:Hankel
2073:Cauchy
1998:Matrix
1939:
1916:
1873:
1854:
75:×
2790:Gamma
2754:Tutte
2616:Shear
2329:Shift
2319:Pauli
2268:Walsh
2178:Moore
2058:Block
1781:Notes
464:= 0,
46:is a
2596:Pick
2566:Gram
2334:Zero
2038:Band
1937:ISBN
1914:ISBN
1871:ISBN
1852:ISBN
1741:The
524:and
457:For
435:For
369:>
292:and
269:and
254:and
182:>
151:<
99:and
52:band
38:, a
2685:Hat
2418:or
2401:or
1890:doi
564:or
556:In
342:if
89:i,j
42:or
30:In
2930::
1931:,
1904:;
1760:.
1734:.
1683:66
1664:56
1652:55
1638:46
1626:45
1614:44
1600:35
1588:34
1576:33
1562:24
1550:23
1538:22
1524:13
1512:12
1500:11
1448:66
1429:56
1417:55
1389:46
1377:45
1365:44
1343:35
1331:34
1319:33
1293:24
1281:23
1269:22
1239:13
1227:12
1215:11
1158:66
1146:65
1132:56
1120:55
1108:54
1094:45
1082:44
1070:43
1056:34
1044:33
1032:32
1018:23
1006:22
994:21
980:12
968:11
914:66
902:65
868:56
856:55
844:54
810:45
798:44
786:43
752:34
740:33
728:32
694:23
682:22
670:21
636:12
624:11
536:A
488:=
471:=
442:=
421:=
400:=
384:.
231:0.
142:if
106::
83:=(
2815:S
2273:Z
1990:e
1983:t
1976:v
1946:.
1923:.
1897:.
1892::
1880:.
1861:.
1832:.
1706:.
1701:]
1695:0
1690:0
1679:A
1671:0
1660:A
1648:A
1634:A
1622:A
1610:A
1596:A
1584:A
1572:A
1558:A
1546:A
1534:A
1520:A
1508:A
1496:A
1489:[
1461:.
1456:]
1444:A
1425:A
1413:A
1405:m
1402:y
1399:s
1385:A
1373:A
1361:A
1350:0
1339:A
1327:A
1315:A
1289:A
1277:A
1265:A
1256:0
1246:0
1235:A
1223:A
1211:A
1204:[
1176:.
1171:]
1165:0
1154:B
1142:B
1128:B
1116:B
1104:B
1090:B
1078:B
1066:B
1052:B
1040:B
1028:B
1014:B
1002:B
990:B
976:B
964:B
958:0
952:[
922:]
910:B
898:B
892:0
877:0
864:B
852:B
840:B
817:0
806:B
794:B
782:B
748:B
736:B
724:B
718:0
690:B
678:B
666:B
658:0
643:0
632:B
620:B
613:[
497:2
494:k
490:n
486:1
483:k
473:n
469:2
466:k
462:1
459:k
447:2
444:k
440:1
437:k
426:2
423:k
419:1
416:k
405:2
402:k
398:1
395:k
372:k
365:|
361:j
355:i
351:|
330:0
327:=
322:j
319:,
316:i
312:a
301:k
297:2
294:k
290:1
287:k
259:2
256:k
252:1
249:k
223:2
219:k
215:,
210:1
206:k
201:;
196:2
192:k
188:+
185:i
179:j
165:1
161:k
154:i
148:j
136:0
133:=
128:j
125:,
122:i
118:a
104:2
101:k
97:1
94:k
85:a
81:A
77:n
73:n
20:)
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