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143:
is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is the interior of the set of all non-negative matrices. While such matrices are commonly found, the term "positive matrix" is only occasionally used due to the possible confusion with
206:: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus the inverse of a positive matrix is not positive or even non-negative, as positive matrices are not monomial, for dimension
134:
70:
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502:
1374:
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1059:
85:
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156:
155:
A rectangular non-negative matrix can be approximated by a decomposition with two other non-negative matrices via
1094:
641:
858:
495:
163:
933:
44:
1089:
611:
1193:
1064:
978:
202:
The inverse of a non-negative matrix is usually not non-negative. The exception is the non-negative
148:, which are different. A matrix which is both non-negative and is positive semidefinite is called a
1298:
1188:
896:
576:
226:
145:
24:
221:
There are a number of groups of matrices that form specializations of non-negative matrices, e.g.
1333:
1262:
1144:
1004:
601:
488:
20:
450:
1203:
786:
591:
351:
327:
176:
1149:
886:
736:
731:
566:
541:
536:
195:
is a non-negative matrix. If the non-singular M-matrix is also symmetric then it is called a
1343:
701:
531:
511:
76:
8:
1364:
1338:
916:
721:
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457:. Springer Series in Computational Mathematics. Vol. 27. Springer. pp. 31–62.
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1208:
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1178:
1139:
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963:
958:
943:
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871:
766:
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661:
631:
626:
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596:
556:
466:
434:
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386:
369:
359:
339:
315:
305:
273:
222:
203:
189:
1421:
1389:
1318:
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1252:
1232:
1168:
1074:
1044:
1029:
1009:
948:
901:
876:
866:
837:
756:
751:
726:
656:
636:
546:
526:
458:
446:
416:
335:
290:
265:
230:
196:
1014:
1119:
1054:
1034:
1019:
999:
983:
881:
812:
802:
761:
646:
616:
462:
294:
1379:
1323:
1303:
1288:
1247:
1124:
1084:
1049:
973:
912:
891:
832:
822:
807:
741:
686:
676:
671:
581:
242:
400:
373:
162:
Eigenvalues and eigenvectors of square positive matrices are described by the
1441:
1384:
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1104:
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827:
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438:
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343:
319:
269:
179:
and every row and column sum/product of a nonnegative matrix is nonnegative.
1267:
1224:
1129:
842:
781:
691:
571:
420:
300:
Horn, R.A.; Johnson, C.R. (2013). "8. Positive and nonnegative matrices".
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681:
551:
31:
1160:
621:
358:. Sigma Series in Applied Mathematics. Vol. 5. Helderman Verlag.
79:
in which all the elements are equal to or greater than zero, that is,
1394:
968:
1328:
192:
480:
350:
415:. Springer Series in Statistics (2nd ed.). Springer.
356:
Positive Linear
Systems: The method of positive operators
88:
47:
326:
128:
64:
262:Nonnegative Matrices in the Mathematical Sciences
129:{\displaystyle x_{ij}\geq 0\qquad \forall {i,j}.}
1439:
286:
255:
496:
304:(2nd ed.). Cambridge University Press.
299:
1070:Fundamental (linear differential equation)
503:
489:
354:; Lifshits, Je.A.; Sobolev, A.V. (1990).
332:Positive Solutions of Operator Equations
1375:Matrix representation of conic sections
413:Non-negative matrices and Markov chains
1440:
407:
484:
445:
289:, 2. Nonnegative Matrices pp. 26–62.
380:
65:{\displaystyle \mathbf {X} \geq 0,}
13:
510:
216:
109:
14:
1459:
157:non-negative matrix factorization
1409:
49:
16:Matrix whose elements are all ≥0
1277:Used in science and engineering
248:
108:
520:Explicitly constrained entries
1:
1294:Fundamental (computer vision)
169:
183:
7:
1060:Duplication and elimination
859:eigenvalues or eigenvectors
463:10.1007/978-3-642-05156-2_2
295:10.1137/1.9781611971262.ch2
236:
10:
1464:
993:With specific applications
622:Discrete Fourier Transform
287:Berman & Plemmons 1994
150:doubly non-negative matrix
146:positive-definite matrices
18:
1403:
1352:
1284:Cabibbo–Kobayashi–Maskawa
1276:
1222:
1158:
992:
911:Satisfying conditions on
910:
856:
795:
519:
455:Matrix Iterative Analysis
227:doubly stochastic matrix
164:Perron–Frobenius theorem
25:Positive-definite matrix
19:Not to be confused with
642:Generalized permutation
270:10.1137/1.9781611971262
21:Totally positive matrix
1416:Mathematics portal
451:"Nonnegative Matrices"
130:
66:
421:10.1007/0-387-32792-4
381:Minc, Henryk (1988).
352:Krasnosel'skii, M. A.
328:Krasnosel'skii, M. A.
233:non-negative matrix.
131:
67:
383:Nonnegative matrices
86:
45:
1365:Linear independence
612:Diagonally dominant
258:Plemmons, Robert J.
188:The inverse of any
1370:Matrix exponential
1360:Jordan normal form
1194:Fisher information
1065:Euclidean distance
979:Totally unimodular
126:
62:
36:nonnegative matrix
1435:
1434:
1427:Category:Matrices
1299:Fuzzy associative
1189:Doubly stochastic
897:Positive-definite
577:Block tridiagonal
472:978-3-642-05156-2
430:978-0-387-29765-1
311:978-1-139-78203-6
256:Berman, Abraham;
223:stochastic matrix
204:monomial matrices
1455:
1422:List of matrices
1414:
1413:
1390:Row echelon form
1334:State transition
1263:Seidel adjacency
1145:Totally positive
1005:Alternating sign
602:Complex Hadamard
505:
498:
491:
482:
481:
476:
442:
404:
377:
347:
338:: P. Noordhoff.
323:
283:
212:
197:Stieltjes matrix
135:
133:
132:
127:
122:
101:
100:
71:
69:
68:
63:
52:
1463:
1462:
1458:
1457:
1456:
1454:
1453:
1452:
1438:
1437:
1436:
1431:
1408:
1399:
1348:
1272:
1218:
1154:
988:
906:
852:
791:
592:Centrosymmetric
515:
509:
479:
473:
431:
393:
366:
312:
302:Matrix Analysis
280:
251:
239:
219:
217:Specializations
207:
186:
172:
141:positive matrix
112:
93:
89:
87:
84:
83:
48:
46:
43:
42:
28:
17:
12:
11:
5:
1461:
1451:
1450:
1433:
1432:
1430:
1429:
1424:
1419:
1404:
1401:
1400:
1398:
1397:
1392:
1387:
1382:
1380:Perfect matrix
1377:
1372:
1367:
1362:
1356:
1354:
1350:
1349:
1347:
1346:
1341:
1336:
1331:
1326:
1321:
1316:
1311:
1306:
1301:
1296:
1291:
1286:
1280:
1278:
1274:
1273:
1271:
1270:
1265:
1260:
1255:
1250:
1245:
1240:
1235:
1229:
1227:
1220:
1219:
1217:
1216:
1211:
1206:
1201:
1196:
1191:
1186:
1181:
1176:
1171:
1165:
1163:
1156:
1155:
1153:
1152:
1150:Transformation
1147:
1142:
1137:
1132:
1127:
1122:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1082:
1077:
1072:
1067:
1062:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1007:
1002:
996:
994:
990:
989:
987:
986:
981:
976:
971:
966:
961:
956:
951:
946:
941:
936:
927:
921:
919:
908:
907:
905:
904:
899:
894:
889:
887:Diagonalizable
884:
879:
874:
869:
863:
861:
857:Conditions on
854:
853:
851:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
799:
797:
793:
792:
790:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
737:Skew-symmetric
734:
732:Skew-Hermitian
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
619:
614:
609:
604:
599:
594:
589:
584:
579:
574:
569:
567:Block-diagonal
564:
559:
554:
549:
544:
542:Anti-symmetric
539:
537:Anti-Hermitian
534:
529:
523:
521:
517:
516:
508:
507:
500:
493:
485:
478:
477:
471:
443:
429:
405:
391:
378:
364:
348:
324:
310:
297:
284:
278:
252:
250:
247:
246:
245:
243:Metzler matrix
238:
235:
218:
215:
185:
182:
181:
180:
171:
168:
137:
136:
125:
121:
118:
115:
111:
107:
104:
99:
96:
92:
73:
72:
61:
58:
55:
51:
15:
9:
6:
4:
3:
2:
1460:
1449:
1446:
1445:
1443:
1428:
1425:
1423:
1420:
1418:
1417:
1412:
1406:
1405:
1402:
1396:
1393:
1391:
1388:
1386:
1385:Pseudoinverse
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1357:
1355:
1353:Related terms
1351:
1345:
1344:Z (chemistry)
1342:
1340:
1337:
1335:
1332:
1330:
1327:
1325:
1322:
1320:
1317:
1315:
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1285:
1282:
1281:
1279:
1275:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1230:
1228:
1226:
1221:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1167:
1166:
1164:
1162:
1157:
1151:
1148:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
997:
995:
991:
985:
982:
980:
977:
975:
972:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
940:
937:
935:
931:
928:
926:
923:
922:
920:
918:
914:
909:
903:
900:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
870:
868:
865:
864:
862:
860:
855:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
800:
798:
794:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
702:Pentadiagonal
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
585:
583:
580:
578:
575:
573:
570:
568:
565:
563:
560:
558:
555:
553:
550:
548:
545:
543:
540:
538:
535:
533:
532:Anti-diagonal
530:
528:
525:
524:
522:
518:
513:
506:
501:
499:
494:
492:
487:
486:
483:
474:
468:
464:
460:
456:
452:
448:
444:
440:
436:
432:
426:
422:
418:
414:
410:
406:
402:
398:
394:
392:0-471-83966-3
388:
384:
379:
375:
371:
367:
365:3-88538-405-1
361:
357:
353:
349:
345:
341:
337:
333:
329:
325:
321:
317:
313:
307:
303:
298:
296:
292:
288:
285:
281:
279:0-89871-321-8
275:
271:
267:
263:
259:
254:
253:
244:
241:
240:
234:
232:
228:
224:
214:
210:
205:
200:
198:
194:
191:
178:
174:
173:
167:
165:
160:
158:
153:
151:
147:
142:
123:
119:
116:
113:
105:
102:
97:
94:
90:
82:
81:
80:
78:
59:
56:
53:
41:
40:
39:
37:
33:
26:
22:
1407:
1339:Substitution
1225:graph theory
722:Quaternionic
712:Persymmetric
696:
454:
412:
382:
355:
331:
301:
261:
249:Bibliography
220:
208:
201:
190:non-singular
187:
161:
154:
149:
140:
138:
74:
35:
29:
1314:Hamiltonian
1238:Biadjacency
1174:Correlation
1090:Householder
1040:Commutation
777:Vandermonde
772:Tridiagonal
707:Permutation
697:Nonnegative
682:Matrix unit
562:Bisymmetric
447:Varga, R.S.
32:mathematics
1214:Transition
1209:Stochastic
1179:Covariance
1161:statistics
1140:Symplectic
1135:Similarity
964:Unimodular
959:Orthogonal
944:Involutory
939:Invertible
934:Projection
930:Idempotent
872:Convergent
767:Triangular
717:Polynomial
662:Hessenberg
632:Equivalent
627:Elementary
607:Copositive
597:Conference
557:Bidiagonal
409:Seneta, E.
401:1150971811
374:1409010096
170:Properties
38:, written
1395:Wronskian
1319:Irregular
1309:Gell-Mann
1258:Laplacian
1253:Incidence
1233:Adjacency
1204:Precision
1169:Centering
1075:Generator
1045:Confusion
1030:Circulant
1010:Augmented
969:Unipotent
949:Nilpotent
925:Congruent
902:Stieltjes
877:Defective
867:Companion
838:Redheffer
757:Symmetric
752:Sylvester
727:Signature
657:Hermitian
637:Frobenius
547:Arrowhead
527:Alternant
439:209916821
385:. Wiley.
344:609079647
336:Groningen
320:817562427
231:symmetric
184:Inversion
110:∀
103:≥
54:≥
1448:Matrices
1442:Category
1223:Used in
1159:Used in
1120:Rotation
1095:Jacobian
1055:Distance
1035:Cofactor
1020:Carleman
1000:Adjugate
984:Weighing
917:inverses
913:products
882:Definite
813:Identity
803:Exchange
796:Constant
762:Toeplitz
647:Hadamard
617:Diagonal
449:(2009).
411:(1981).
330:(1964).
264:. SIAM.
260:(1994).
237:See also
193:M-matrix
1324:Overlap
1289:Density
1248:Edmonds
1125:Seifert
1085:Hessian
1050:Coxeter
974:Unitary
892:Hurwitz
823:Of ones
808:Hilbert
742:Skyline
687:Metzler
677:Logical
672:Integer
582:Boolean
514:classes
1243:Degree
1184:Design
1115:Random
1105:Payoff
1100:Moment
1025:Cartan
1015:BĂ©zout
954:Normal
828:Pascal
818:Lehmer
747:Sparse
667:Hollow
652:Hankel
587:Cauchy
512:Matrix
469:
437:
427:
399:
389:
372:
362:
342:
318:
308:
276:
211:> 1
77:matrix
1304:Gamma
1268:Tutte
1130:Shear
843:Shift
833:Pauli
782:Walsh
692:Moore
572:Block
177:trace
75:is a
1110:Pick
1080:Gram
848:Zero
552:Band
467:ISBN
435:OCLC
425:ISBN
397:OCLC
387:ISBN
370:OCLC
360:ISBN
340:OCLC
316:OCLC
306:ISBN
274:ISBN
175:The
34:, a
23:and
1199:Hat
932:or
915:or
459:doi
417:doi
291:doi
266:doi
199:.
30:In
1444::
465:.
453:.
433:.
423:.
395:.
368:.
334:.
314:.
272:.
229:;
225:;
213:.
166:.
159:.
152:.
139:A
1329:S
787:Z
504:e
497:t
490:v
475:.
461::
441:.
419::
403:.
376:.
346:.
322:.
293::
282:.
268::
209:n
124:.
120:j
117:,
114:i
106:0
98:j
95:i
91:x
60:,
57:0
50:X
27:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.