1529:
1565:
247:
85:
481:
283:
308:
because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of
1187:
477:
447:
443:
401:
1401:
620:
171:
1492:
1606:
459:
1411:
1177:
37:
588:
555:
497:
425:
1212:
759:
976:
613:
332:
309:
1051:
350:
98:
161:
except for those on the main diagonal, which are unconstrained. That is, a
Metzler matrix is any matrix
1207:
729:
1599:
1311:
1182:
1096:
1416:
1306:
1014:
694:
1451:
1380:
1262:
1122:
719:
606:
102:
1321:
904:
709:
371:
286:
574:
1630:
1267:
1004:
854:
849:
684:
659:
654:
255:
578:
1592:
1572:
1461:
819:
649:
629:
345:
138:
28:
31:
in which all the off-diagonal components are nonnegative (equal to or greater than zero):
8:
1482:
1456:
1034:
839:
829:
312:
are always
Metzler matrices, and that probability distributions are always non-negative.
1580:
1533:
1487:
1477:
1431:
1426:
1355:
1291:
1157:
894:
889:
824:
814:
679:
405:
305:
301:
106:
1625:
1544:
1528:
1331:
1326:
1316:
1296:
1257:
1252:
1081:
1076:
1061:
1056:
1047:
1042:
989:
884:
834:
779:
749:
744:
724:
714:
674:
584:
561:
551:
531:
503:
493:
465:
455:
431:
421:
381:
1539:
1507:
1436:
1375:
1370:
1350:
1286:
1192:
1162:
1147:
1127:
1066:
1019:
994:
984:
955:
874:
869:
844:
774:
754:
664:
644:
543:
523:
515:
413:
386:
1132:
1237:
1172:
1152:
1137:
1117:
1101:
999:
930:
920:
879:
764:
734:
1576:
1497:
1441:
1421:
1406:
1365:
1242:
1202:
1167:
1091:
1030:
1009:
950:
940:
925:
859:
794:
789:
699:
376:
134:
565:
535:
527:
507:
1619:
1502:
1360:
1301:
1232:
1222:
1217:
1142:
1071:
945:
935:
864:
784:
769:
704:
469:
435:
91:
417:
1385:
1342:
1247:
960:
899:
809:
689:
485:
158:
1227:
1197:
965:
799:
669:
316:
130:
20:
1278:
739:
1512:
1086:
1446:
365:
360:
355:
598:
548:
Introduction to
Dynamic Systems: Theory, Modes & Applications
320:
323:
because of the corresponding property for nonnegative matrices.
105:. Their properties can be derived by applying the properties of
1564:
242:{\displaystyle A=(a_{ij});\quad a_{ij}\geq 0,\quad i\neq j.}
16:
Square matrix whose off-diagonal entries are nonnegative
575:"§3.4 Matrices with the Minkowski or Metzler Property"
258:
174:
40:
492:. Pure and Applied Mathematics. Wiley Interscience.
252:Metzler matrices are also sometimes referred to as
476:
277:
241:
79:
410:Nonnegative Matrices in the Mathematical Sciences
292:is equivalent to a negated quasipositive matrix.
97:Metzler matrices appear in stability analysis of
80:{\displaystyle \forall _{i\neq j}\,x_{ij}\geq 0.}
1617:
452:Positive Linear Systems: Theory and Applications
400:
1600:
614:
442:
124:
304:of a Metzler (or quasipositive) matrix is a
1607:
1593:
1188:Fundamental (linear differential equation)
621:
607:
572:
542:
490:Nonnegative Matrices in Dynamical Systems
90:It is named after the American economist
57:
573:Kemp, Murray C.; Kimura, Yoshio (1978).
514:
1493:Matrix representation of conic sections
1618:
580:Introduction to Mathematical Economics
602:
1559:
326:
368:, a specific kind of Metzler matrix
99:time delayed differential equations
13:
628:
42:
14:
1642:
1563:
1527:
1395:Used in science and engineering
393:
226:
203:
638:Explicitly constrained entries
583:. Springer. pp. 102–114.
270:
264:
197:
181:
1:
1412:Fundamental (computer vision)
310:continuous-time Markov chains
295:
1579:. You can help Knowledge by
7:
1178:Duplication and elimination
977:eigenvalues or eigenvectors
351:Delay differential equation
338:
157:if all of its elements are
10:
1647:
1558:
1111:With specific applications
740:Discrete Fourier Transform
520:Positive 1D and 2D Systems
125:Definition and terminology
1521:
1470:
1402:Cabibbo–Kobayashi–Maskawa
1394:
1340:
1276:
1110:
1029:Satisfying conditions on
1028:
974:
913:
637:
550:. Wiley. pp. 204–6.
528:10.1007/978-1-4471-0221-2
333:Perron–Frobenius theorem
315:A Metzler matrix has an
109:to matrices of the form
103:linear dynamical systems
760:Generalized permutation
418:10.1137/1.9781611971262
278:{\displaystyle Z^{(-)}}
155:essentially nonnegative
1534:Mathematics portal
454:. Wiley Interscience.
279:
243:
81:
280:
244:
121:is a Metzler matrix.
82:
346:Nonnegative matrices
256:
172:
107:nonnegative matrices
38:
1571:This article about
1483:Linear independence
730:Diagonally dominant
406:Plemmons, Robert J.
319:in the nonnegative
1488:Matrix exponential
1478:Jordan normal form
1312:Fisher information
1183:Euclidean distance
1097:Totally unimodular
306:nonnegative matrix
275:
239:
77:
1588:
1587:
1553:
1552:
1545:Category:Matrices
1417:Fuzzy associative
1307:Doubly stochastic
1015:Positive-definite
695:Block tridiagonal
544:Luenberger, David
516:Kaczorek, Tadeusz
461:978-1-118-03127-8
382:Stochastic matrix
327:Relevant theorems
1638:
1609:
1602:
1595:
1567:
1560:
1540:List of matrices
1532:
1531:
1508:Row echelon form
1452:State transition
1381:Seidel adjacency
1263:Totally positive
1123:Alternating sign
720:Complex Hadamard
623:
616:
609:
600:
599:
594:
569:
539:
511:
482:Neumann, Michael
473:
439:
387:Positive systems
285:-matrices, as a
284:
282:
281:
276:
274:
273:
248:
246:
245:
240:
216:
215:
196:
195:
165:which satisfies
86:
84:
83:
78:
70:
69:
56:
55:
1646:
1645:
1641:
1640:
1639:
1637:
1636:
1635:
1616:
1615:
1614:
1613:
1556:
1554:
1549:
1526:
1517:
1466:
1390:
1336:
1272:
1106:
1024:
970:
909:
710:Centrosymmetric
633:
627:
597:
591:
558:
500:
478:Berman, Abraham
462:
448:Rinaldi, Sergio
444:Farina, Lorenzo
428:
402:Berman, Abraham
396:
391:
341:
329:
298:
263:
259:
257:
254:
253:
208:
204:
188:
184:
173:
170:
169:
127:
62:
58:
45:
41:
39:
36:
35:
17:
12:
11:
5:
1644:
1634:
1633:
1628:
1612:
1611:
1604:
1597:
1589:
1586:
1585:
1568:
1551:
1550:
1548:
1547:
1542:
1537:
1522:
1519:
1518:
1516:
1515:
1510:
1505:
1500:
1498:Perfect matrix
1495:
1490:
1485:
1480:
1474:
1472:
1468:
1467:
1465:
1464:
1459:
1454:
1449:
1444:
1439:
1434:
1429:
1424:
1419:
1414:
1409:
1404:
1398:
1396:
1392:
1391:
1389:
1388:
1383:
1378:
1373:
1368:
1363:
1358:
1353:
1347:
1345:
1338:
1337:
1335:
1334:
1329:
1324:
1319:
1314:
1309:
1304:
1299:
1294:
1289:
1283:
1281:
1274:
1273:
1271:
1270:
1268:Transformation
1265:
1260:
1255:
1250:
1245:
1240:
1235:
1230:
1225:
1220:
1215:
1210:
1205:
1200:
1195:
1190:
1185:
1180:
1175:
1170:
1165:
1160:
1155:
1150:
1145:
1140:
1135:
1130:
1125:
1120:
1114:
1112:
1108:
1107:
1105:
1104:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1064:
1059:
1054:
1045:
1039:
1037:
1026:
1025:
1023:
1022:
1017:
1012:
1007:
1005:Diagonalizable
1002:
997:
992:
987:
981:
979:
975:Conditions on
972:
971:
969:
968:
963:
958:
953:
948:
943:
938:
933:
928:
923:
917:
915:
911:
910:
908:
907:
902:
897:
892:
887:
882:
877:
872:
867:
862:
857:
855:Skew-symmetric
852:
850:Skew-Hermitian
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
685:Block-diagonal
682:
677:
672:
667:
662:
660:Anti-symmetric
657:
655:Anti-Hermitian
652:
647:
641:
639:
635:
634:
626:
625:
618:
611:
603:
596:
595:
589:
570:
556:
540:
512:
498:
474:
460:
440:
426:
397:
395:
392:
390:
389:
384:
379:
377:Hurwitz matrix
374:
369:
363:
358:
353:
348:
342:
340:
337:
336:
335:
328:
325:
297:
294:
272:
269:
266:
262:
250:
249:
238:
235:
232:
229:
225:
222:
219:
214:
211:
207:
202:
199:
194:
191:
187:
183:
180:
177:
151:quasi-positive
135:linear algebra
126:
123:
88:
87:
76:
73:
68:
65:
61:
54:
51:
48:
44:
25:Metzler matrix
15:
9:
6:
4:
3:
2:
1643:
1632:
1629:
1627:
1624:
1623:
1621:
1610:
1605:
1603:
1598:
1596:
1591:
1590:
1584:
1582:
1578:
1574:
1569:
1566:
1562:
1561:
1557:
1546:
1543:
1541:
1538:
1536:
1535:
1530:
1524:
1523:
1520:
1514:
1511:
1509:
1506:
1504:
1503:Pseudoinverse
1501:
1499:
1496:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1475:
1473:
1471:Related terms
1469:
1463:
1462:Z (chemistry)
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1405:
1403:
1400:
1399:
1397:
1393:
1387:
1384:
1382:
1379:
1377:
1374:
1372:
1369:
1367:
1364:
1362:
1359:
1357:
1354:
1352:
1349:
1348:
1346:
1344:
1339:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1310:
1308:
1305:
1303:
1300:
1298:
1295:
1293:
1290:
1288:
1285:
1284:
1282:
1280:
1275:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1201:
1199:
1196:
1194:
1191:
1189:
1186:
1184:
1181:
1179:
1176:
1174:
1171:
1169:
1166:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1115:
1113:
1109:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1049:
1046:
1044:
1041:
1040:
1038:
1036:
1032:
1027:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
982:
980:
978:
973:
967:
964:
962:
959:
957:
954:
952:
949:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
918:
916:
912:
906:
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
820:Pentadiagonal
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
650:Anti-diagonal
648:
646:
643:
642:
640:
636:
631:
624:
619:
617:
612:
610:
605:
604:
601:
592:
590:0-387-90304-6
586:
582:
581:
576:
571:
567:
563:
559:
557:0-471-02594-1
553:
549:
545:
541:
537:
533:
529:
525:
521:
517:
513:
509:
505:
501:
499:0-471-62074-2
495:
491:
487:
486:Stern, Ronald
483:
479:
475:
471:
467:
463:
457:
453:
449:
445:
441:
437:
433:
429:
427:0-89871-321-8
423:
419:
415:
411:
407:
403:
399:
398:
388:
385:
383:
380:
378:
375:
373:
370:
367:
364:
362:
359:
357:
354:
352:
349:
347:
344:
343:
334:
331:
330:
324:
322:
318:
313:
311:
307:
303:
293:
291:
289:
267:
260:
236:
233:
230:
227:
223:
220:
217:
212:
209:
205:
200:
192:
189:
185:
178:
175:
168:
167:
166:
164:
160:
156:
152:
148:
147:quasipositive
144:
140:
136:
133:, especially
132:
122:
120:
116:
113: +
112:
108:
104:
101:and positive
100:
95:
93:
92:Lloyd Metzler
74:
71:
66:
63:
59:
52:
49:
46:
34:
33:
32:
30:
26:
22:
1631:Matrix stubs
1581:expanding it
1570:
1555:
1525:
1457:Substitution
1343:graph theory
840:Quaternionic
830:Persymmetric
804:
579:
547:
522:. Springer.
519:
489:
451:
409:
394:Bibliography
314:
299:
287:
251:
162:
159:non-negative
154:
150:
146:
142:
128:
118:
114:
110:
96:
89:
24:
18:
1432:Hamiltonian
1356:Biadjacency
1292:Correlation
1208:Householder
1158:Commutation
895:Vandermonde
890:Tridiagonal
825:Permutation
815:Nonnegative
800:Matrix unit
680:Bisymmetric
317:eigenvector
302:exponential
131:mathematics
21:mathematics
1620:Categories
1332:Transition
1327:Stochastic
1297:Covariance
1279:statistics
1258:Symplectic
1253:Similarity
1082:Unimodular
1077:Orthogonal
1062:Involutory
1057:Invertible
1052:Projection
1048:Idempotent
990:Convergent
885:Triangular
835:Polynomial
780:Hessenberg
750:Equivalent
745:Elementary
725:Copositive
715:Conference
675:Bidiagonal
566:1422165904
536:1050930884
508:1409010310
296:Properties
141:is called
1513:Wronskian
1437:Irregular
1427:Gell-Mann
1376:Laplacian
1371:Incidence
1351:Adjacency
1322:Precision
1287:Centering
1193:Generator
1163:Confusion
1148:Circulant
1128:Augmented
1087:Unipotent
1067:Nilpotent
1043:Congruent
1020:Stieltjes
995:Defective
985:Companion
956:Redheffer
875:Symmetric
870:Sylvester
845:Signature
775:Hermitian
755:Frobenius
665:Arrowhead
645:Alternant
470:815646165
450:(2011) .
436:722474576
268:−
231:≠
218:≥
72:≥
50:≠
43:∀
1626:Matrices
1573:matrices
1341:Used in
1277:Used in
1238:Rotation
1213:Jacobian
1173:Distance
1153:Cofactor
1138:Carleman
1118:Adjugate
1102:Weighing
1035:inverses
1031:products
1000:Definite
931:Identity
921:Exchange
914:Constant
880:Toeplitz
765:Hadamard
735:Diagonal
546:(1979).
518:(2002).
488:(1989).
412:. SIAM.
408:(1994).
372:Z-matrix
366:Q-matrix
361:P-matrix
356:M-matrix
339:See also
117:, where
1442:Overlap
1407:Density
1366:Edmonds
1243:Seifert
1203:Hessian
1168:Coxeter
1092:Unitary
1010:Hurwitz
941:Of ones
926:Hilbert
860:Skyline
805:Metzler
795:Logical
790:Integer
700:Boolean
632:classes
321:orthant
290:-matrix
143:Metzler
1361:Degree
1302:Design
1233:Random
1223:Payoff
1218:Moment
1143:Cartan
1133:BĂ©zout
1072:Normal
946:Pascal
936:Lehmer
865:Sparse
785:Hollow
770:Hankel
705:Cauchy
630:Matrix
587:
564:
554:
534:
506:
496:
468:
458:
434:
424:
139:matrix
29:matrix
1575:is a
1422:Gamma
1386:Tutte
1248:Shear
961:Shift
951:Pauli
900:Walsh
810:Moore
690:Block
153:) or
27:is a
1577:stub
1228:Pick
1198:Gram
966:Zero
670:Band
585:ISBN
562:OCLC
552:ISBN
532:OCLC
504:OCLC
494:ISBN
466:OCLC
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