1255:
997:
1077:
728:
212:
1313:
856:
541:
350:
423:
828:
1250:{\displaystyle \det P=\prod _{i=1}^{n}\lambda _{i}\leq {\bigg (}{1 \over n}\sum _{i=1}^{n}\lambda _{i}{\bigg )}^{n}=\left({1 \over n}\operatorname {tr} P\right)^{n}=1^{n}=1,}
595:
1068:
761:. By dividing each column by its length, it can be seen that the result is equivalent to the special case where each column has length 1, in other words if
1520:
135:
1539:
1493:
1266:
992:{\displaystyle \left|\det N\right|={\bigg (}\prod _{i=1}^{n}\|v_{i}\|{\bigg )}\left|\det M\right|\leq \prod _{i=1}^{n}\|v_{i}\|.}
31:
480:
282:
560:
1384:
365:
737:
is less than or equal to the product of its diagonal entries. Sometimes this is also known as
Hadamard's inequality.
1419:
The result is sometimes stated in terms of row vectors. That this is equivalent is seen by applying the transpose.
791:
734:
1364:
1056:
54:
1598:
1593:
758:
62:
578:
1003:
1512:
1550:
1535:
1516:
1489:
1470:
Proof follows, with minor modifications, the second proof given in Maz'ya & Shaposhnikova.
437:
723:{\displaystyle \det(P)=\det(N)^{2}\leq \prod _{i=1}^{n}\|v_{i}\|^{2}=\prod _{i=1}^{n}p_{ii}.}
1504:
1451:
1336:
50:
1344:
750:
586:
82:
1553:
1340:
847:
550:
448:
222:
66:
1587:
1505:
429:
17:
769:
74:
58:
38:
1456:
1439:
1026:
226:
1558:
238:
1355:
are an orthogonal set. Many other proofs can be found in the literature.
436:
for which equality holds, i.e. those with orthogonal columns, are called
70:
78:
221:
vectors are non-zero, equality in
Hadamard's inequality is achieved
207:{\displaystyle \left|\det(N)\right|\leq \prod _{i=1}^{n}\|v_{i}\|.}
1308:{\displaystyle \left|\det M\right|={\sqrt {\det P}}\leq 1.}
846:
and equality is achieved if and only if the vectors are an
1438:
Różański, Michał; Wituła, Roman; Hetmaniok, Edyta (2017).
536:{\displaystyle |\operatorname {det} (N)|\leq n^{n/2}.}
345:{\displaystyle \left|\det(N)\right|\leq B^{n}n^{n/2}.}
1575:
Beckenbach, Edwin F; Bellman, Richard Ernest (1965).
1269:
1080:
859:
794:
598:
546:
Equality in this bound is attained for a real matrix
483:
368:
285:
138:
118:Specifically, Hadamard's inequality states that if
1307:
1249:
991:
822:
722:
535:
417:
344:
206:
69:in terms of the lengths of its column vectors. In
1440:"More subtle versions of the Hadamard inequality"
1175:
1126:
925:
881:
418:{\displaystyle \left|\det(N)\right|\leq n^{n/2}.}
1385:"Hadamard theorem - Encyclopedia of Mathematics"
1291:
1275:
1081:
935:
865:
800:
614:
599:
374:
291:
144:
1484:Maz'ya, Vladimir; Shaposhnikova, T. O. (1999).
1530:Riesz, Frigyes; Szőkefalvi-Nagy, Béla (1990).
8:
1507:Inequalities: A Journey into Linear Analysis
1069:inequality of arithmetic and geometric means
983:
970:
920:
907:
671:
657:
198:
185:
1486:Jacques Hadamard: A Universal Mathematician
108:in terms of the lengths of these vectors ||
1351:are an orthonormal set and the columns of
823:{\displaystyle \left|\det M\right|\leq 1,}
1455:
1289:
1268:
1232:
1219:
1195:
1180:
1174:
1173:
1166:
1156:
1145:
1131:
1125:
1124:
1115:
1105:
1094:
1079:
977:
964:
953:
924:
923:
914:
901:
890:
880:
879:
858:
793:
708:
698:
687:
674:
664:
651:
640:
627:
597:
520:
516:
504:
484:
482:
474:. Then Hadamard's inequality states that
402:
398:
367:
329:
325:
315:
284:
192:
179:
168:
137:
1406:
1404:
1376:
1327:'s must all be equal and their sum is
1318:If there is equality then each of the
587:Decomposition of a semidefinite matrix
1047:. Since the length of each column of
7:
1331:, so they must all be 1. The matrix
1051:is 1, each entry in the diagonal of
785:
745:The result is trivial if the matrix
1444:Linear Algebra and Its Applications
850:. The general result now follows:
47:Hadamard's theorem on determinants
25:
355:In particular, if the entries of
49:) is a result first published by
455:, whose entries are bounded by |
1347:—in other words the columns of
233:Alternate forms and corollaries
1021:is the conjugate transpose of
624:
617:
608:
602:
505:
501:
495:
485:
383:
377:
300:
294:
153:
147:
1:
443:More generally, suppose that
241:is that if the entries of an
122:is the matrix having columns
561:positive-semidefinite matrix
30:Not to be confused with the
753:, so assume the columns of
32:Hermite–Hadamard inequality
1615:
1503:Garling, D. J. H. (2007).
1410:Maz'ya & Shaposhnikova
73:terms, when restricted to
29:
1457:10.1016/j.laa.2017.07.003
776:is the matrix having the
733:So, the determinant of a
89:dimensions marked out by
735:positive definite matrix
359:are +1 and −1 only then
1579:. Springer. p. 64.
1554:"Hadamard's Inequality"
1488:. AMS. pp. 383ff.
462: | ≤ 1, for each
1534:. Dover. p. 176.
1389:encyclopediaofmath.org
1309:
1251:
1161:
1110:
993:
969:
906:
824:
724:
703:
656:
556:is a Hadamard matrix.
537:
419:
346:
208:
184:
1511:. Cambridge. p.
1310:
1252:
1141:
1090:
994:
949:
886:
825:
725:
683:
636:
538:
420:
347:
209:
164:
43:Hadamard's inequality
1365:Fischer's inequality
1267:
1078:
857:
792:
759:linearly independent
596:
481:
366:
283:
136:
1532:Functional Analysis
579:conjugate transpose
18:Hadamard inequality
1551:Weisstein, Eric W.
1305:
1247:
989:
820:
720:
566:can be written as
533:
415:
342:
204:
65:whose entries are
1522:978-0-521-69973-0
1297:
1203:
1139:
844:
843:
438:Hadamard matrices
53:in 1893. It is a
16:(Redirected from
1606:
1580:
1564:
1563:
1545:
1526:
1510:
1499:
1471:
1468:
1462:
1461:
1459:
1435:
1429:
1426:
1420:
1417:
1411:
1408:
1399:
1398:
1396:
1395:
1381:
1314:
1312:
1311:
1306:
1298:
1290:
1285:
1281:
1256:
1254:
1253:
1248:
1237:
1236:
1224:
1223:
1218:
1214:
1204:
1196:
1185:
1184:
1179:
1178:
1171:
1170:
1160:
1155:
1140:
1132:
1130:
1129:
1120:
1119:
1109:
1104:
998:
996:
995:
990:
982:
981:
968:
963:
945:
941:
929:
928:
919:
918:
905:
900:
885:
884:
875:
871:
838:
829:
827:
826:
821:
810:
806:
786:
783:as columns then
729:
727:
726:
721:
716:
715:
702:
697:
679:
678:
669:
668:
655:
650:
632:
631:
542:
540:
539:
534:
529:
528:
524:
508:
488:
451:matrix of order
424:
422:
421:
416:
411:
410:
406:
390:
386:
351:
349:
348:
343:
338:
337:
333:
320:
319:
307:
303:
225:the vectors are
213:
211:
210:
205:
197:
196:
183:
178:
160:
156:
77:, it bounds the
51:Jacques Hadamard
21:
1614:
1613:
1609:
1608:
1607:
1605:
1604:
1603:
1584:
1583:
1574:
1571:
1569:Further reading
1549:
1548:
1542:
1529:
1523:
1502:
1496:
1483:
1480:
1475:
1474:
1469:
1465:
1437:
1436:
1432:
1427:
1423:
1418:
1414:
1409:
1402:
1393:
1391:
1383:
1382:
1378:
1373:
1361:
1345:identity matrix
1343:, so it is the
1326:
1274:
1270:
1265:
1264:
1228:
1194:
1190:
1189:
1172:
1162:
1111:
1076:
1075:
1067:. Applying the
1046:
1040:
1036:
973:
934:
930:
910:
864:
860:
855:
854:
836:
799:
795:
790:
789:
781:
766:
743:
704:
670:
660:
623:
594:
593:
512:
479:
478:
460:
394:
373:
369:
364:
363:
321:
311:
290:
286:
281:
280:
262:
253:are bounded by
235:
188:
143:
139:
134:
133:
127:
113:
98:
83:Euclidean space
67:complex numbers
45:(also known as
35:
28:
23:
22:
15:
12:
11:
5:
1612:
1610:
1602:
1601:
1596:
1586:
1585:
1582:
1581:
1570:
1567:
1566:
1565:
1546:
1540:
1527:
1521:
1500:
1494:
1479:
1476:
1473:
1472:
1463:
1430:
1421:
1412:
1400:
1375:
1374:
1372:
1369:
1368:
1367:
1360:
1357:
1341:diagonalizable
1322:
1316:
1315:
1304:
1301:
1296:
1293:
1288:
1284:
1280:
1277:
1273:
1258:
1257:
1246:
1243:
1240:
1235:
1231:
1227:
1222:
1217:
1213:
1210:
1207:
1202:
1199:
1193:
1188:
1183:
1177:
1169:
1165:
1159:
1154:
1151:
1148:
1144:
1138:
1135:
1128:
1123:
1118:
1114:
1108:
1103:
1100:
1097:
1093:
1089:
1086:
1083:
1042:
1038:
1034:
1025:, and let the
1000:
999:
988:
985:
980:
976:
972:
967:
962:
959:
956:
952:
948:
944:
940:
937:
933:
927:
922:
917:
913:
909:
904:
899:
896:
893:
889:
883:
878:
874:
870:
867:
863:
848:orthogonal set
842:
841:
832:
830:
819:
816:
813:
809:
805:
802:
798:
779:
764:
742:
739:
731:
730:
719:
714:
711:
707:
701:
696:
693:
690:
686:
682:
677:
673:
667:
663:
659:
654:
649:
646:
643:
639:
635:
630:
626:
622:
619:
616:
613:
610:
607:
604:
601:
551:if and only if
544:
543:
532:
527:
523:
519:
515:
511:
507:
503:
500:
497:
494:
491:
487:
470:between 1 and
458:
426:
425:
414:
409:
405:
401:
397:
393:
389:
385:
382:
379:
376:
372:
353:
352:
341:
336:
332:
328:
324:
318:
314:
310:
306:
302:
299:
296:
293:
289:
260:
234:
231:
223:if and only if
215:
214:
203:
200:
195:
191:
187:
182:
177:
174:
171:
167:
163:
159:
155:
152:
149:
146:
142:
125:
111:
96:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1611:
1600:
1597:
1595:
1592:
1591:
1589:
1578:
1573:
1572:
1568:
1561:
1560:
1555:
1552:
1547:
1543:
1541:0-486-66289-6
1537:
1533:
1528:
1524:
1518:
1514:
1509:
1508:
1501:
1497:
1495:0-8218-1923-2
1491:
1487:
1482:
1481:
1477:
1467:
1464:
1458:
1453:
1449:
1445:
1441:
1434:
1431:
1425:
1422:
1416:
1413:
1407:
1405:
1401:
1390:
1386:
1380:
1377:
1370:
1366:
1363:
1362:
1358:
1356:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1325:
1321:
1302:
1299:
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1286:
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1278:
1271:
1263:
1262:
1261:
1244:
1241:
1238:
1233:
1229:
1225:
1220:
1215:
1211:
1208:
1205:
1200:
1197:
1191:
1186:
1181:
1167:
1163:
1157:
1152:
1149:
1146:
1142:
1136:
1133:
1121:
1116:
1112:
1106:
1101:
1098:
1095:
1091:
1087:
1084:
1074:
1073:
1072:
1070:
1066:
1062:
1058:
1055:is 1, so the
1054:
1050:
1045:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1005:
986:
978:
974:
965:
960:
957:
954:
950:
946:
942:
938:
931:
915:
911:
902:
897:
894:
891:
887:
876:
872:
868:
861:
853:
852:
851:
849:
840:
833:
831:
817:
814:
811:
807:
803:
796:
788:
787:
784:
782:
775:
771:
767:
760:
756:
752:
748:
740:
738:
736:
717:
712:
709:
705:
699:
694:
691:
688:
684:
680:
675:
665:
661:
652:
647:
644:
641:
637:
633:
628:
620:
611:
605:
592:
591:
590:
588:
584:
580:
576:
572:
569:
565:
562:
557:
555:
552:
549:
530:
525:
521:
517:
513:
509:
498:
492:
489:
477:
476:
475:
473:
469:
465:
461:
454:
450:
446:
441:
439:
435:
431:
430:combinatorics
412:
407:
403:
399:
395:
391:
387:
380:
370:
362:
361:
360:
358:
339:
334:
330:
326:
322:
316:
312:
308:
304:
297:
287:
279:
278:
277:
275:
271:
267:
263:
256:
252:
248:
244:
240:
232:
230:
228:
224:
220:
201:
193:
189:
180:
175:
172:
169:
165:
161:
157:
150:
140:
132:
131:
130:
128:
121:
116:
114:
107:
103:
99:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
1599:Determinants
1594:Inequalities
1577:Inequalities
1576:
1557:
1531:
1506:
1485:
1466:
1447:
1443:
1433:
1424:
1415:
1392:. Retrieved
1388:
1379:
1352:
1348:
1339:, therefore
1332:
1328:
1323:
1319:
1317:
1259:
1064:
1060:
1052:
1048:
1043:
1030:
1022:
1018:
1014:
1010:
1006:
1001:
845:
834:
777:
773:
770:unit vectors
762:
754:
746:
744:
732:
582:
577:denotes the
574:
570:
567:
563:
558:
553:
547:
545:
471:
467:
463:
456:
452:
444:
442:
433:
427:
356:
354:
273:
269:
265:
258:
254:
250:
246:
242:
236:
218:
216:
123:
119:
117:
109:
105:
101:
94:
90:
86:
75:real numbers
46:
42:
36:
1450:: 500–511.
1027:eigenvalues
1009:, consider
432:, matrices
264: | ≤
115: ||.
71:geometrical
59:determinant
39:mathematics
1588:Categories
1478:References
1394:2020-06-15
227:orthogonal
1559:MathWorld
1337:Hermitian
1300:≤
1209:
1164:λ
1143:∑
1122:≤
1113:λ
1092:∏
984:‖
971:‖
951:∏
947:≤
921:‖
908:‖
888:∏
812:≤
685:∏
672:‖
658:‖
638:∏
634:≤
510:≤
493:
392:≤
309:≤
239:corollary
199:‖
186:‖
166:∏
162:≤
1359:See also
751:singular
589:). Then
573:, where
268:for all
129:, then
100:for 1 ≤
93:vectors
1428:Garling
449:complex
276:, then
249:matrix
217:If the
57:on the
27:Theorem
1538:
1519:
1492:
1017:where
257:, so |
79:volume
63:matrix
1371:Notes
1057:trace
1041:, … λ
1004:prove
741:Proof
585:(see
447:is a
61:of a
55:bound
1536:ISBN
1517:ISBN
1490:ISBN
1033:be λ
772:and
768:are
757:are
272:and
1513:233
1452:doi
1448:532
1335:is
1292:det
1276:det
1260:so
1082:det
1063:is
1059:of
1037:, λ
1029:of
1007:(1)
1002:To
936:det
866:det
801:det
749:is
615:det
600:det
581:of
490:det
428:In
375:det
292:det
245:by
145:det
85:of
81:in
37:In
1590::
1556:.
1515:.
1446:.
1442:.
1403:^
1387:.
1303:1.
1206:tr
1071:,
1015:MM
559:A
466:,
459:ij
440:.
261:ij
237:A
229:.
104:≤
41:,
1562:.
1544:.
1525:.
1498:.
1460:.
1454::
1397:.
1353:N
1349:M
1333:P
1329:n
1324:i
1320:λ
1295:P
1287:=
1283:|
1279:M
1272:|
1245:,
1242:1
1239:=
1234:n
1230:1
1226:=
1221:n
1216:)
1212:P
1201:n
1198:1
1192:(
1187:=
1182:n
1176:)
1168:i
1158:n
1153:1
1150:=
1147:i
1137:n
1134:1
1127:(
1117:i
1107:n
1102:1
1099:=
1096:i
1088:=
1085:P
1065:n
1061:P
1053:P
1049:M
1044:n
1039:2
1035:1
1031:P
1023:M
1019:M
1013:=
1011:P
987:.
979:i
975:v
966:n
961:1
958:=
955:i
943:|
939:M
932:|
926:)
916:i
912:v
903:n
898:1
895:=
892:i
882:(
877:=
873:|
869:N
862:|
839:)
837:1
835:(
818:,
815:1
808:|
804:M
797:|
780:i
778:e
774:M
765:i
763:e
755:N
747:N
718:.
713:i
710:i
706:p
700:n
695:1
692:=
689:i
681:=
676:2
666:i
662:v
653:n
648:1
645:=
642:i
629:2
625:)
621:N
618:(
612:=
609:)
606:P
603:(
583:N
575:N
571:N
568:N
564:P
554:N
548:N
531:.
526:2
522:/
518:n
514:n
506:|
502:)
499:N
496:(
486:|
472:n
468:j
464:i
457:N
453:n
445:N
434:N
413:.
408:2
404:/
400:n
396:n
388:|
384:)
381:N
378:(
371:|
357:N
340:.
335:2
331:/
327:n
323:n
317:n
313:B
305:|
301:)
298:N
295:(
288:|
274:j
270:i
266:B
259:N
255:B
251:N
247:n
243:n
219:n
202:.
194:i
190:v
181:n
176:1
173:=
170:i
158:|
154:)
151:N
148:(
141:|
126:i
124:v
120:N
112:i
110:v
106:n
102:i
97:i
95:v
91:n
87:n
34:.
20:)
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