1832:
786:
4210:
The specialization of the above section to skewfields was a consequence of the application to projective geometry, and not intrinsic to the nature of sesquilinear forms. Only the minor modifications needed to take into account the non-commutativity of multiplication are required to generalize the
398:
Conventions differ as to which argument should be linear. In the commutative case, we shall take the first to be linear, as is common in the mathematical literature, except in the section devoted to sesquilinear forms on complex vector spaces. There we use the other convention and take the first
1615:
3035:
587:
2675:. More specific terminology then also applies: the division ring is a field, the anti-automorphism is also an automorphism, and the right module is a vector space. The following applies to a left module with suitable reordering of expressions.
4418:
2858:
3989:
3598:
2212:
276:
3408:
4498:
1955:
1072:
3534:
1400:
966:
4785:
592:
579:
2939:
2547:
2097:
2476:
2029:
1561:
449:
1827:{\displaystyle \varphi (w,z)=\varphi \left(\sum _{i}w_{i}e_{i},\sum _{j}z_{j}e_{j}\right)=\sum _{i}\sum _{j}{\overline {w_{i}}}z_{j}\varphi \left(e_{i},e_{j}\right)=w^{\dagger }Az.}
1236:
1141:
870:
781:{\displaystyle {\begin{aligned}&\varphi (x+y,z+w)=\varphi (x,z)+\varphi (x,w)+\varphi (y,z)+\varphi (y,w)\\&\varphi (ax,by)={\overline {a}}b\,\varphi (x,y)\end{aligned}}}
313:
3267:
2126:
3226:
2583:
993:
897:
2651:
2380:
2269:
1859:
833:
1484:
1460:
1440:
1288:
484:
In the more general noncommutative setting, with right modules we take the second argument to be linear and with left modules we take the first argument to be linear.
2421:
350:
1100:
479:
2308:
1420:
1328:
1195:
2131:
1610:
1584:
1266:
1020:
920:
1883:
1516:
1308:
1165:
539:
376:
4292:
71:
from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector.
2781:
3860:
3540:
106:
in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any
5287:
399:
argument to be conjugate-linear (i.e. antilinear) and the second to be linear. This is the convention used mostly by physicists and originates in
382:, defined more precisely, below. There is no particular reason to restrict the definition to the complex numbers; it can be defined for arbitrary
195:
1033:
1337:
930:
5064:
3341:
5505:
2481:
4424:
2034:
5398:
5167:. In all cases, alternating bilinear forms are a subset of skew-symmetric bilinear forms, and need not be considered separately.
1888:
5343:
5220:
4138:
projective geometries correspond to the nondegenerate sesquilinear forms on the underlying vector space. A sesquilinear form
3494:
2128:
is given (again, using the "physics" convention of linearity in the second and conjugate linearity in the first variable) by
4725:
4185:
To achieve full generality of this statement, and since every desarguesian projective geometry may be coordinatized by a
544:
3030:{\displaystyle W^{\perp }=\{\mathbf {v} \in M\mid \varphi (\mathbf {v} ,\mathbf {w} )=0,\ \forall \mathbf {w} \in W\}.}
5082:
Proceedings of the NATO Advanced Study
Institute, Held at Nijenrode Castle, Breukelen, the Netherlands, 8–20 July 1974
5522:
5325:
5299:
4131:
4202:. (In the geometric literature these are still referred to as either left or right vector spaces over skewfields.)
2598:
2327:
2441:
1994:
996:
17:
1524:
5470:
5369:
4193:
extended the definition of a sesquilinear form to a division ring, which requires replacing vector spaces by
413:
1200:
1105:
5622:
5537:
5475:
3272:
That is, a sesquilinear form is reflexive precisely when the derived orthogonality relation is symmetric.
838:
5391:
5364:
5105:
284:
5578:
5491:
5359:
5249:
5122:
3231:
2102:
3619:
3190:
2555:
971:
875:
352:
This product may be generalized to situations where one is not working with an orthonormal basis for
4806:. A Hermitian form is necessarily reflexive, and if it is nonzero, the associated antiautomorphism
5598:
5527:
5155:
4813:
4009:
2228:
1242:
54:
in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a
1837:
794:
5627:
5101:
1490:. Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.
5496:
5465:
5384:
4951:, which has the same underlying set and the same addition, but whose multiplication operation (
1519:
5234:
Baer's terminology gives a third way to refer to these ideas, so he must be read with caution.
5091:
1469:
1445:
1425:
1273:
5547:
5501:
5444:
5128:
2589:
2397:
322:
1079:
454:
5430:
5309:
4237:
4194:
2281:
1970:
may also refer to a different concept than that explained below: it may refer to a certain
1587:
1405:
1313:
1173:
519:
403:
390:, informally understood to be a generalized concept of "complex conjugation" for the ring.
168:
75:
68:
8:
5426:
4413:{\displaystyle \varphi (x+y,z+w)=\varphi (x,z)+\varphi (x,w)+\varphi (y,z)+\varphi (y,w)}
4070:
1862:
1331:
118:
107:
1592:
1566:
1248:
1002:
902:
5280:
5194:
4221:
3121:
1975:
1868:
1501:
1293:
1150:
1023:
524:
383:
361:
147:
111:
5449:
5421:
5339:
5321:
5295:
5216:
4923:
2726:
1971:
1144:
407:
316:
99:
5562:
5439:
5186:
4247:
2853:{\displaystyle \varphi (x\alpha ,y\beta )=\sigma (\alpha )\,\varphi (x,y)\,\beta .}
2672:
2654:
2318:
1239:
387:
98:
that is linear in one argument and "twists" the linearity of the other argument by
59:
5542:
5305:
5291:
4042:
3984:{\displaystyle \varphi (x,y)=x_{1}y_{1}{}^{q}+x_{2}y_{2}{}^{q}+x_{3}y_{3}{}^{q}.}
3117:
2705:
47:
410:. It is also consistent with the definition of the usual (Euclidean) product of
62:
4058:
3645:
3593:{\displaystyle \sigma (\sigma (\alpha ))=\varepsilon \alpha \varepsilon ^{-1}.}
2550:
2391:
2387:
1027:
507:
493:
55:
5616:
5435:
5407:
4942:
4190:
4186:
4134:. A result of Birkhoff and von Neumann (1936) shows that the correlations of
2735:
2219:
2215:
180:
176:
122:
67:
meaning "one and a half". The basic concept of the dot product – producing a
39:
2207:{\displaystyle \langle w,z\rangle =\sum _{i=1}^{n}{\overline {w}}_{i}z_{i}.}
271:{\displaystyle \langle w,z\rangle =\sum _{i=1}^{n}{\overline {w}}_{i}z_{i}.}
4135:
3769:
3703:
925:
5557:
4079:
3789:
2383:
497:
43:
31:
5177:
Birkhoff, G.; von
Neumann, J. (1936), "The logic of quantum mechanics",
131:, and this means that the "vectors" should be replaced by elements of a
5198:
4062:
1168:
511:
400:
103:
51:
5085:
2902:
5190:
140:. In a very general setting, sesquilinear forms can be defined over
3641:
3403:{\displaystyle \varphi (x,y)=\sigma (\varphi (y,x))\,\varepsilon .}
1518:
is a finite-dimensional complex vector space, then relative to any
132:
5163:, skew-symmetric and symmetric bilinear forms coincide since then
5376:
2272:
172:
163:
Sesquilinear forms abstract and generalize the basic notion of a
2588:
The matrix representation of a complex skew-Hermitian form is a
1030:
these are in one-to-one correspondence with complex linear maps
5044:
4211:
arbitrary field version of the definition to arbitrary rings.
2386:. One can show that a complex sesquilinear form is Hermitian
358:, or even any basis at all. By inserting an extra factor of
4493:{\displaystyle \varphi (cx,dy)=c\,\varphi (x,y)\,\sigma (d)}
924:
A complex sesquilinear form can also be viewed as a complex
2317:
The matrix representation of a complex
Hermitian form is a
2595:
A complex skew-Hermitian form applied to a single vector
1950:{\displaystyle A_{ij}:=\varphi \left(e_{i},e_{j}\right).}
1067:{\displaystyle {\overline {V}}\otimes V\to \mathbb {C} .}
42:
that, in turn, is a generalization of the concept of the
2549:
Every complex skew-Hermitian form can be written as the
3529:{\displaystyle \sigma (\varepsilon )=\varepsilon ^{-1}}
1395:{\displaystyle \psi (w,z)={\overline {\varphi (z,w)}}.}
961:{\displaystyle {\overline {V}}\times V\to \mathbb {C} }
58:
manner, thus the name; which originates from the Latin
2665:
This section applies unchanged when the division ring
74:
A motivating special case is a sesquilinear form on a
4887:
is a bilinear form. In particular, if, in this case,
4728:
4427:
4295:
3863:
3543:
3497:
3344:
3234:
3193:
2942:
2784:
2601:
2558:
2484:
2444:
2400:
2330:
2284:
2231:
2134:
2105:
2037:
1997:
1891:
1871:
1840:
1618:
1595:
1569:
1527:
1504:
1472:
1448:
1428:
1408:
1340:
1316:
1296:
1276:
1251:
1203:
1176:
1153:
1108:
1082:
1036:
1005:
974:
933:
905:
878:
841:
797:
590:
547:
527:
457:
416:
364:
325:
287:
198:
4780:{\displaystyle \varphi (x,y)=\sigma (\varphi (y,x))}
2324:
A complex
Hermitian form applied to a single vector
5176:
4969:, where the product on the right is the product in
5279:
4779:
4492:
4412:
3983:
3792:. With respect to the standard basis we can write
3669:-Hermitian form is reflexive, and every reflexive
3592:
3528:
3402:
3261:
3220:
3029:
2852:
2645:
2577:
2541:
2470:
2415:
2374:
2302:
2263:
2206:
2120:
2091:
2023:
1949:
1877:
1853:
1826:
1604:
1578:
1555:
1510:
1478:
1454:
1434:
1414:
1394:
1322:
1302:
1282:
1260:
1230:
1189:
1159:
1135:
1094:
1066:
1014:
987:
960:
914:
891:
864:
827:
780:
574:{\displaystyle \varphi :V\times V\to \mathbb {C} }
573:
533:
473:
443:
370:
344:
307:
270:
2225:A minus sign is introduced in the Hermitian form
1310:we can define a second complex sesquilinear form
183:. In such cases, the standard Hermitian form on
5614:
5288:Ergebnisse der Mathematik und ihrer Grenzgebiete
5244:Faure, Claude-Alain; Frölicher, Alfred (2000),
3768:be the three dimensional vector space over the
5243:
5119:Finite Geometry and Combinatorial Applications
4087:of the subspaces that inverts inclusion, i.e.
5392:
5315:
4593:. This relation need not be symmetric, i.e.
2542:{\displaystyle s(w,z)=-{\overline {s(z,w)}}.}
1466:. If they are negatives of one another, then
1442:will be different. If they are the same then
3021:
2956:
2147:
2135:
2092:{\displaystyle h(w,z)={\overline {h(z,w)}}.}
211:
199:
5116:
4975:. It follows from this that a right (left)
2471:{\displaystyle s:V\times V\to \mathbb {C} }
2024:{\displaystyle h:V\times V\to \mathbb {C} }
5399:
5385:
4057:: In this section, sesquilinear forms are
506:: In this section, sesquilinear forms are
5277:
5141:
4477:
4458:
4065:) in their second (resp. first) argument.
4048:
3476:-Hermitian form, it follows that for all
3393:
2843:
2824:
2464:
2108:
2017:
1057:
954:
855:
755:
567:
431:
5333:
5261:
5137:
5135:
4905:is a vector space with a bilinear form.
3101:can be inferred from the context), when
1586:a sesquilinear form is represented by a
1556:{\displaystyle \left\{e_{i}\right\}_{i}}
1493:
487:
158:
4205:
3640:. The fixed points of this map form a
3275:
2660:
444:{\displaystyle w,z\in \mathbb {C} ^{n}}
171:. Hermitian forms are commonly seen in
14:
5615:
5213:Linear Algebra and Projective Geometry
4562:with respect to the sesquilinear form
2425:
1231:{\displaystyle w\mapsto \varphi (w,z)}
1136:{\displaystyle w\mapsto \varphi (z,w)}
121:requires that the scalars come from a
5380:
5132:
2278:A vector space with a Hermitian form
899:is the complex conjugate of a scalar
5316:Gruenberg, K.W.; Weir, A.J. (1977),
5210:
1270:Given any complex sesquilinear form
865:{\displaystyle a,b\in \mathbb {C} .}
5532:
308:{\displaystyle {\overline {w}}_{i}}
24:
5406:
4987:can be turned into a left (right)
3007:
2869:for any nonzero sesquilinear form
2438:), is a complex sesquilinear form
378:into the product, one obtains the
25:
5639:
5523:Compact operator on Hilbert space
5352:
5017:can be viewed as a bilinear form
2863:The associated anti-automorphism
1959:
27:Generalization of a bilinear form
3453:is implied, respectively simply
3262:{\displaystyle \varphi (y,x)=0.}
3146:
3011:
2988:
2980:
2960:
2884:
2121:{\displaystyle \mathbb {C} ^{n}}
5255:
5237:
4041:associated to this form is the
4030:-sesquilinear form. The matrix
3221:{\displaystyle \varphi (x,y)=0}
2578:{\displaystyle i:={\sqrt {-1}}}
2436:antisymmetric sesquilinear form
2099:The standard Hermitian form on
1865:. The components of the matrix
988:{\displaystyle {\overline {V}}}
892:{\displaystyle {\overline {a}}}
110:and the twist is provided by a
5228:
5204:
5170:
5147:
5110:
5095:
5073:
5057:
4999:. Thus, the sesquilinear form
4819:Since for an antiautomorphism
4774:
4771:
4759:
4753:
4744:
4732:
4487:
4481:
4474:
4462:
4449:
4431:
4407:
4395:
4386:
4374:
4365:
4353:
4344:
4332:
4323:
4299:
3879:
3867:
3562:
3559:
3553:
3547:
3507:
3501:
3390:
3387:
3375:
3369:
3360:
3348:
3250:
3238:
3209:
3197:
3153:
2992:
2976:
2840:
2828:
2821:
2815:
2806:
2788:
2646:{\displaystyle |z|_{s}=s(z,z)}
2640:
2628:
2612:
2603:
2527:
2515:
2500:
2488:
2460:
2375:{\displaystyle |z|_{h}=h(z,z)}
2369:
2357:
2341:
2332:
2297:
2285:
2077:
2065:
2053:
2041:
2013:
1634:
1622:
1380:
1368:
1356:
1344:
1225:
1213:
1207:
1130:
1118:
1112:
1053:
997:complex conjugate vector space
950:
771:
759:
736:
718:
707:
695:
686:
674:
665:
653:
644:
632:
623:
599:
563:
13:
1:
5334:Jacobson, Nathan J. (2009) ,
5290:, Band 44, Berlin, New York:
5271:
2678:
2264:{\displaystyle ww^{*}-zz^{*}}
393:
4045:. This is a Hermitian form.
3739:the bilinear form is called
2531:
2180:
2081:
1854:{\displaystyle w^{\dagger }}
1751:
1384:
1042:
980:
939:
884:
828:{\displaystyle x,y,z,w\in V}
747:
510:in their first argument and
294:
244:
7:
5365:Encyclopedia of Mathematics
5106:Encyclopedia of Mathematics
5038:
1989:symmetric sesquilinear form
10:
5644:
5492:Hilbert projection theorem
5320:(2nd ed.), Springer,
5250:Kluwer Academic Publishers
5246:Modern Projective Geometry
5123:Cambridge University Press
3757:
2889:Given a sesquilinear form
2875:is uniquely determined by
1991:), is a sesquilinear form
491:
5571:
5515:
5484:
5471:Cauchy–Schwarz inequality
5458:
5414:
5278:Dembowski, Peter (1968),
4922:can also be viewed as an
3696:In the special case that
38:is a generalization of a
5211:Baer, Reinhold (2005) ,
5050:
4881:must be commutative and
2585:times a Hermitian form.
1479:{\displaystyle \varphi }
1455:{\displaystyle \varphi }
1435:{\displaystyle \varphi }
1283:{\displaystyle \varphi }
1167:(i.e. an element of the
5338:(2nd ed.), Dover,
3725:is a bilinear form and
2416:{\displaystyle z\in V.}
345:{\displaystyle w_{i}~.}
146:-modules for arbitrary
4781:
4494:
4414:
4049:In projective geometry
3985:
3675:-sesquilinear form is
3594:
3530:
3404:
3263:
3222:
3031:
2854:
2647:
2579:
2543:
2472:
2417:
2376:
2304:
2265:
2208:
2173:
2122:
2093:
2025:
1951:
1879:
1855:
1828:
1606:
1580:
1557:
1512:
1480:
1456:
1436:
1416:
1396:
1324:
1304:
1284:
1262:
1232:
1191:
1161:
1137:
1096:
1095:{\displaystyle z\in V}
1068:
1016:
989:
962:
916:
893:
866:
829:
782:
575:
535:
475:
474:{\displaystyle w^{*}z}
445:
372:
346:
309:
272:
237:
102:(referred to as being
5502:Polarization identity
5445:Orthogonal complement
5179:Annals of Mathematics
4893:is a skewfield, then
4782:
4495:
4415:
3986:
3603:It also follows that
3595:
3531:
3420:, the form is called
3405:
3264:
3223:
3032:
2919:orthogonal complement
2855:
2648:
2590:skew-Hermitian matrix
2580:
2544:
2473:
2418:
2377:
2305:
2303:{\displaystyle (V,h)}
2266:
2222:is a Hermitian form.
2209:
2153:
2123:
2094:
2026:
1952:
1880:
1856:
1829:
1607:
1581:
1558:
1513:
1494:Matrix representation
1481:
1457:
1437:
1417:
1415:{\displaystyle \psi }
1397:
1325:
1323:{\displaystyle \psi }
1305:
1285:
1263:
1233:
1197:). Likewise, the map
1192:
1190:{\displaystyle V^{*}}
1162:
1138:
1097:
1069:
1017:
990:
963:
917:
894:
867:
830:
783:
576:
536:
488:Complex vector spaces
476:
446:
373:
347:
310:
273:
217:
159:Informal introduction
50:. A bilinear form is
5476:Riesz representation
5431:L-semi-inner product
5117:Simeon Ball (2015),
4908:An antiautomorphism
4726:
4691:A sesquilinear form
4616:A sesquilinear form
4425:
4293:
4206:Over arbitrary rings
3861:
3687:-Hermitian for some
3541:
3495:
3342:
3276:Hermitian variations
3232:
3191:
3158:A sesquilinear form
2940:
2782:
2661:Over a division ring
2599:
2556:
2482:
2442:
2398:
2328:
2282:
2271:to define the group
2229:
2214:More generally, the
2132:
2103:
2035:
1995:
1889:
1869:
1838:
1616:
1593:
1567:
1525:
1502:
1470:
1446:
1426:
1406:
1338:
1314:
1294:
1274:
1249:
1201:
1174:
1151:
1106:
1080:
1034:
1003:
972:
931:
903:
876:
839:
795:
588:
545:
525:
520:complex vector space
455:
414:
362:
323:
285:
196:
169:complex vector space
76:complex vector space
5623:Functional analysis
5497:Parseval's identity
5466:Bessel's inequality
5360:"Sesquilinear form"
4816:(i.e. of order 2).
4556:to another element
4071:projective geometry
3848:and define the map
3319:such that, for all
3286:-sesquilinear form
2743:such that, for all
2725:with an associated
2653:is always a purely
2432:skew-Hermitian form
2426:Skew-Hermitian form
1863:conjugate transpose
1332:conjugate transpose
581:is sesquilinear if
380:skew-Hermitian form
119:projective geometry
100:complex conjugation
4777:
4490:
4410:
4108:for all subspaces
3981:
3590:
3526:
3400:
3259:
3218:
3147:§ Reflexivity
3027:
2850:
2690:-sesquilinear form
2643:
2575:
2539:
2468:
2413:
2372:
2300:
2261:
2204:
2118:
2089:
2021:
1976:Hermitian manifold
1947:
1875:
1851:
1824:
1738:
1728:
1690:
1657:
1605:{\displaystyle A,}
1602:
1579:{\displaystyle V,}
1576:
1553:
1508:
1476:
1452:
1432:
1412:
1392:
1320:
1300:
1280:
1261:{\displaystyle V.}
1258:
1228:
1187:
1157:
1133:
1092:
1064:
1024:universal property
1015:{\displaystyle V.}
1012:
985:
958:
915:{\displaystyle a.}
912:
889:
862:
825:
778:
776:
571:
531:
471:
441:
368:
342:
305:
268:
117:An application in
112:field automorphism
5610:
5609:
5553:Sesquilinear form
5506:Parallelogram law
5450:Orthonormal basis
5345:978-0-486-47189-1
5282:Finite geometries
5222:978-0-486-44565-6
5102:Sesquilinear form
5080:"Combinatorics",
4175:(if and) only if
3006:
2727:anti-automorphism
2573:
2534:
2183:
2084:
1972:differential form
1878:{\displaystyle A}
1754:
1729:
1719:
1681:
1648:
1511:{\displaystyle V}
1387:
1303:{\displaystyle V}
1160:{\displaystyle V}
1145:linear functional
1045:
983:
942:
887:
750:
534:{\displaystyle V}
408:quantum mechanics
371:{\displaystyle i}
338:
317:complex conjugate
297:
247:
36:sesquilinear form
16:(Redirected from
5635:
5440:Prehilbert space
5401:
5394:
5387:
5378:
5377:
5373:
5348:
5330:
5312:
5285:
5265:
5259:
5253:
5252:
5241:
5235:
5232:
5226:
5225:
5208:
5202:
5201:
5174:
5168:
5166:
5162:
5151:
5145:
5139:
5130:
5126:
5114:
5108:
5099:
5093:
5089:
5077:
5071:
5061:
5034:
5016:
4998:
4992:
4986:
4980:
4974:
4968:
4955:) is defined by
4954:
4950:
4940:
4934:
4921:
4904:
4898:
4892:
4886:
4880:
4874:
4867:
4861:
4851:
4824:
4811:
4805:
4799:
4786:
4784:
4783:
4778:
4718:
4713:if there exists
4708:
4687:
4681:
4671:
4656:
4633:
4612:
4602:
4592:
4577:
4567:
4561:
4551:
4542:
4536:
4526:
4520:
4499:
4497:
4496:
4491:
4419:
4417:
4416:
4411:
4283:
4276:
4255:
4248:antiautomorphism
4245:
4235:
4229:
4219:
4199:
4181:
4174:
4168:
4162:
4143:
4125:
4119:
4113:
4107:
4086:
4077:
4040:
4029:
4023:
4017:
4012:automorphism of
4007:
3990:
3988:
3987:
3982:
3977:
3976:
3971:
3968:
3967:
3958:
3957:
3945:
3944:
3939:
3936:
3935:
3926:
3925:
3913:
3912:
3907:
3904:
3903:
3894:
3893:
3853:
3847:
3819:
3787:
3781:
3767:
3749:
3738:
3731:
3724:
3719:is commutative,
3718:
3712:
3701:
3692:
3686:
3674:
3668:
3653:
3639:
3617:
3599:
3597:
3596:
3591:
3586:
3585:
3535:
3533:
3532:
3527:
3525:
3524:
3487:
3481:
3475:
3452:
3442:
3436:
3425:
3419:
3409:
3407:
3406:
3401:
3334:
3328:
3318:
3312:
3307:if there exists
3304:
3291:
3285:
3268:
3266:
3265:
3260:
3227:
3225:
3224:
3219:
3183:
3177:
3163:
3143:
3133:
3115:
3100:
3094:
3084:
3069:
3064:with respect to
3063:
3049:
3036:
3034:
3033:
3028:
3014:
3004:
2991:
2983:
2963:
2952:
2951:
2932:
2927:with respect to
2926:
2916:
2910:
2901:and a subspace (
2900:
2894:
2880:
2874:
2868:
2859:
2857:
2856:
2851:
2774:
2768:
2758:
2752:
2742:
2733:
2724:
2703:
2697:
2689:
2670:
2655:imaginary number
2652:
2650:
2649:
2644:
2621:
2620:
2615:
2606:
2584:
2582:
2581:
2576:
2574:
2566:
2548:
2546:
2545:
2540:
2535:
2530:
2510:
2477:
2475:
2474:
2469:
2467:
2434:(also called an
2422:
2420:
2419:
2414:
2394:is real for all
2381:
2379:
2378:
2373:
2350:
2349:
2344:
2335:
2319:Hermitian matrix
2309:
2307:
2306:
2301:
2270:
2268:
2267:
2262:
2260:
2259:
2244:
2243:
2213:
2211:
2210:
2205:
2200:
2199:
2190:
2189:
2184:
2176:
2172:
2167:
2127:
2125:
2124:
2119:
2117:
2116:
2111:
2098:
2096:
2095:
2090:
2085:
2080:
2060:
2030:
2028:
2027:
2022:
2020:
1956:
1954:
1953:
1948:
1943:
1939:
1938:
1937:
1925:
1924:
1904:
1903:
1884:
1882:
1881:
1876:
1860:
1858:
1857:
1852:
1850:
1849:
1833:
1831:
1830:
1825:
1814:
1813:
1801:
1797:
1796:
1795:
1783:
1782:
1765:
1764:
1755:
1750:
1749:
1740:
1737:
1727:
1715:
1711:
1710:
1709:
1700:
1699:
1689:
1677:
1676:
1667:
1666:
1656:
1611:
1609:
1608:
1603:
1585:
1583:
1582:
1577:
1562:
1560:
1559:
1554:
1552:
1551:
1546:
1542:
1541:
1517:
1515:
1514:
1509:
1485:
1483:
1482:
1477:
1461:
1459:
1458:
1453:
1441:
1439:
1438:
1433:
1421:
1419:
1418:
1413:
1401:
1399:
1398:
1393:
1388:
1383:
1363:
1329:
1327:
1326:
1321:
1309:
1307:
1306:
1301:
1289:
1287:
1286:
1281:
1267:
1265:
1264:
1259:
1240:conjugate-linear
1237:
1235:
1234:
1229:
1196:
1194:
1193:
1188:
1186:
1185:
1166:
1164:
1163:
1158:
1142:
1140:
1139:
1134:
1101:
1099:
1098:
1093:
1073:
1071:
1070:
1065:
1060:
1046:
1038:
1021:
1019:
1018:
1013:
994:
992:
991:
986:
984:
976:
967:
965:
964:
959:
957:
943:
935:
921:
919:
918:
913:
898:
896:
895:
890:
888:
880:
871:
869:
868:
863:
858:
834:
832:
831:
826:
787:
785:
784:
779:
777:
751:
743:
713:
594:
580:
578:
577:
572:
570:
540:
538:
537:
532:
514:in their second.
480:
478:
477:
472:
467:
466:
450:
448:
447:
442:
440:
439:
434:
404:bra–ket notation
388:antiautomorphism
377:
375:
374:
369:
357:
351:
349:
348:
343:
336:
335:
334:
314:
312:
311:
306:
304:
303:
298:
290:
277:
275:
274:
269:
264:
263:
254:
253:
248:
240:
236:
231:
188:
154:
145:
137:
130:
97:
84:. This is a map
83:
60:numerical prefix
21:
5643:
5642:
5638:
5637:
5636:
5634:
5633:
5632:
5613:
5612:
5611:
5606:
5599:Segal–Bargmann
5567:
5538:Hilbert–Schmidt
5528:Densely defined
5511:
5480:
5454:
5410:
5405:
5358:
5355:
5346:
5336:Basic Algebra I
5328:
5318:Linear Geometry
5302:
5292:Springer-Verlag
5274:
5269:
5268:
5260:
5256:
5242:
5238:
5233:
5229:
5223:
5209:
5205:
5191:10.2307/1968621
5175:
5171:
5164:
5154:
5152:
5148:
5140:
5133:
5115:
5111:
5100:
5096:
5088:: 456–457, 1975
5079:
5078:
5074:
5062:
5058:
5053:
5041:
5018:
5000:
4994:
4988:
4982:
4976:
4970:
4956:
4952:
4946:
4936:
4926:
4909:
4900:
4899:is a field and
4894:
4888:
4882:
4876:
4869:
4863:
4853:
4826:
4820:
4807:
4801:
4791:
4727:
4724:
4723:
4714:
4692:
4683:
4673:
4658:
4643:
4617:
4604:
4603:does not imply
4594:
4579:
4569:
4563:
4557:
4547:
4538:
4528:
4522:
4504:
4426:
4423:
4422:
4294:
4291:
4290:
4279:
4260:
4251:
4241:
4231:
4225:
4215:
4208:
4195:
4176:
4170:
4164:
4149:
4139:
4121:
4115:
4109:
4091:
4082:
4073:
4051:
4043:identity matrix
4039:
4031:
4025:
4019:
4013:
3995:
3972:
3970:
3969:
3963:
3959:
3953:
3949:
3940:
3938:
3937:
3931:
3927:
3921:
3917:
3908:
3906:
3905:
3899:
3895:
3889:
3885:
3862:
3859:
3858:
3849:
3845:
3838:
3831:
3821:
3817:
3810:
3803:
3793:
3783:
3772:
3763:
3760:
3744:
3733:
3726:
3720:
3714:
3707:
3697:
3688:
3676:
3670:
3658:
3649:
3623:
3604:
3578:
3574:
3542:
3539:
3538:
3517:
3513:
3496:
3493:
3492:
3483:
3477:
3465:
3448:
3438:
3437:, it is called
3431:
3421:
3414:
3343:
3340:
3339:
3330:
3320:
3314:
3308:
3294:
3287:
3281:
3278:
3233:
3230:
3229:
3192:
3189:
3188:
3179:
3169:
3159:
3156:
3135:
3134:does not imply
3125:
3102:
3096:
3086:
3080:
3071:
3065:
3055:
3041:
3010:
2987:
2979:
2959:
2947:
2943:
2941:
2938:
2937:
2928:
2922:
2912:
2906:
2896:
2890:
2887:
2876:
2870:
2864:
2783:
2780:
2779:
2770:
2760:
2754:
2744:
2738:
2729:
2708:
2706:bi-additive map
2699:
2693:
2685:
2681:
2666:
2663:
2616:
2611:
2610:
2602:
2600:
2597:
2596:
2565:
2557:
2554:
2553:
2511:
2509:
2483:
2480:
2479:
2463:
2443:
2440:
2439:
2428:
2399:
2396:
2395:
2390:the associated
2345:
2340:
2339:
2331:
2329:
2326:
2325:
2312:Hermitian space
2283:
2280:
2279:
2255:
2251:
2239:
2235:
2230:
2227:
2226:
2218:on any complex
2195:
2191:
2185:
2175:
2174:
2168:
2157:
2133:
2130:
2129:
2112:
2107:
2106:
2104:
2101:
2100:
2061:
2059:
2036:
2033:
2032:
2016:
1996:
1993:
1992:
1987:(also called a
1962:
1933:
1929:
1920:
1916:
1915:
1911:
1896:
1892:
1890:
1887:
1886:
1870:
1867:
1866:
1845:
1841:
1839:
1836:
1835:
1809:
1805:
1791:
1787:
1778:
1774:
1773:
1769:
1760:
1756:
1745:
1741:
1739:
1733:
1723:
1705:
1701:
1695:
1691:
1685:
1672:
1668:
1662:
1658:
1652:
1647:
1643:
1617:
1614:
1613:
1594:
1591:
1590:
1568:
1565:
1564:
1547:
1537:
1533:
1529:
1528:
1526:
1523:
1522:
1503:
1500:
1499:
1496:
1471:
1468:
1467:
1447:
1444:
1443:
1427:
1424:
1423:
1407:
1404:
1403:
1364:
1362:
1339:
1336:
1335:
1315:
1312:
1311:
1295:
1292:
1291:
1275:
1272:
1271:
1250:
1247:
1246:
1202:
1199:
1198:
1181:
1177:
1175:
1172:
1171:
1152:
1149:
1148:
1107:
1104:
1103:
1081:
1078:
1077:
1056:
1037:
1035:
1032:
1031:
1028:tensor products
1004:
1001:
1000:
975:
973:
970:
969:
953:
934:
932:
929:
928:
904:
901:
900:
879:
877:
874:
873:
854:
840:
837:
836:
796:
793:
792:
775:
774:
742:
711:
710:
591:
589:
586:
585:
566:
546:
543:
542:
526:
523:
522:
500:
490:
462:
458:
456:
453:
452:
435:
430:
429:
415:
412:
411:
396:
363:
360:
359:
353:
330:
326:
324:
321:
320:
299:
289:
288:
286:
283:
282:
259:
255:
249:
239:
238:
232:
221:
197:
194:
193:
184:
161:
150:
141:
133:
126:
85:
79:
48:Euclidean space
28:
23:
22:
15:
12:
11:
5:
5641:
5631:
5630:
5628:Linear algebra
5625:
5608:
5607:
5605:
5604:
5596:
5590:compact &
5575:
5573:
5569:
5568:
5566:
5565:
5560:
5555:
5550:
5545:
5540:
5535:
5533:Hermitian form
5530:
5525:
5519:
5517:
5513:
5512:
5510:
5509:
5499:
5494:
5488:
5486:
5482:
5481:
5479:
5478:
5473:
5468:
5462:
5460:
5456:
5455:
5453:
5452:
5447:
5442:
5433:
5424:
5418:
5416:
5415:Basic concepts
5412:
5411:
5408:Hilbert spaces
5404:
5403:
5396:
5389:
5381:
5375:
5374:
5354:
5353:External links
5351:
5350:
5349:
5344:
5331:
5326:
5313:
5300:
5273:
5270:
5267:
5266:
5254:
5236:
5227:
5221:
5203:
5185:(4): 823–843,
5169:
5146:
5142:Dembowski 1968
5131:
5109:
5094:
5072:
5069:(2007) pg. 255
5065:Anthony Knapp
5063:footnote 1 in
5055:
5054:
5052:
5049:
5048:
5047:
5040:
5037:
4788:
4787:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4640:orthosymmetric
4501:
4500:
4489:
4486:
4483:
4480:
4476:
4473:
4470:
4467:
4464:
4461:
4457:
4454:
4451:
4448:
4445:
4442:
4439:
4436:
4433:
4430:
4420:
4409:
4406:
4403:
4400:
4397:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4298:
4207:
4204:
4128:
4127:
4067:
4066:
4050:
4047:
4035:
3992:
3991:
3980:
3975:
3966:
3962:
3956:
3952:
3948:
3943:
3934:
3930:
3924:
3920:
3916:
3911:
3902:
3898:
3892:
3888:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3843:
3836:
3829:
3815:
3808:
3801:
3759:
3756:
3752:skew-symmetric
3646:additive group
3601:
3600:
3589:
3584:
3581:
3577:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3536:
3523:
3520:
3516:
3512:
3509:
3506:
3503:
3500:
3464:For a nonzero
3459:anti-Hermitian
3445:anti-Hermitian
3411:
3410:
3399:
3396:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3277:
3274:
3270:
3269:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3155:
3152:
3076:
3038:
3037:
3026:
3023:
3020:
3017:
3013:
3009:
3003:
3000:
2997:
2994:
2990:
2986:
2982:
2978:
2975:
2972:
2969:
2966:
2962:
2958:
2955:
2950:
2946:
2895:over a module
2886:
2883:
2861:
2860:
2849:
2846:
2842:
2839:
2836:
2833:
2830:
2827:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2680:
2677:
2662:
2659:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2619:
2614:
2609:
2605:
2572:
2569:
2564:
2561:
2551:imaginary unit
2538:
2533:
2529:
2526:
2523:
2520:
2517:
2514:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2466:
2462:
2459:
2456:
2453:
2450:
2447:
2427:
2424:
2412:
2409:
2406:
2403:
2392:quadratic form
2388:if and only if
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2348:
2343:
2338:
2334:
2299:
2296:
2293:
2290:
2287:
2258:
2254:
2250:
2247:
2242:
2238:
2234:
2203:
2198:
2194:
2188:
2182:
2179:
2171:
2166:
2163:
2160:
2156:
2152:
2149:
2146:
2143:
2140:
2137:
2115:
2110:
2088:
2083:
2079:
2076:
2073:
2070:
2067:
2064:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2019:
2015:
2012:
2009:
2006:
2003:
2000:
1985:Hermitian form
1981:
1980:
1968:Hermitian form
1961:
1960:Hermitian form
1958:
1946:
1942:
1936:
1932:
1928:
1923:
1919:
1914:
1910:
1907:
1902:
1899:
1895:
1874:
1848:
1844:
1823:
1820:
1817:
1812:
1808:
1804:
1800:
1794:
1790:
1786:
1781:
1777:
1772:
1768:
1763:
1759:
1753:
1748:
1744:
1736:
1732:
1726:
1722:
1718:
1714:
1708:
1704:
1698:
1694:
1688:
1684:
1680:
1675:
1671:
1665:
1661:
1655:
1651:
1646:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1601:
1598:
1575:
1572:
1550:
1545:
1540:
1536:
1532:
1507:
1495:
1492:
1489:
1488:skew-Hermitian
1486:is said to be
1475:
1465:
1462:is said to be
1451:
1431:
1411:
1391:
1386:
1382:
1379:
1376:
1373:
1370:
1367:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1319:
1299:
1279:
1257:
1254:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1184:
1180:
1156:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1091:
1088:
1085:
1063:
1059:
1055:
1052:
1049:
1044:
1041:
1011:
1008:
982:
979:
956:
952:
949:
946:
941:
938:
911:
908:
886:
883:
861:
857:
853:
850:
847:
844:
824:
821:
818:
815:
812:
809:
806:
803:
800:
789:
788:
773:
770:
767:
764:
761:
758:
754:
749:
746:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
712:
709:
706:
703:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
593:
569:
565:
562:
559:
556:
553:
550:
530:
516:
515:
494:Antidual space
489:
486:
470:
465:
461:
438:
433:
428:
425:
422:
419:
395:
392:
367:
341:
333:
329:
302:
296:
293:
279:
278:
267:
262:
258:
252:
246:
243:
235:
230:
227:
224:
220:
216:
213:
210:
207:
204:
201:
165:Hermitian form
160:
157:
125:(skew field),
26:
18:Hermitian form
9:
6:
4:
3:
2:
5640:
5629:
5626:
5624:
5621:
5620:
5618:
5603:
5602:
5597:
5595:
5593:
5589:
5585:
5581:
5577:
5576:
5574:
5570:
5564:
5561:
5559:
5556:
5554:
5551:
5549:
5546:
5544:
5541:
5539:
5536:
5534:
5531:
5529:
5526:
5524:
5521:
5520:
5518:
5514:
5507:
5503:
5500:
5498:
5495:
5493:
5490:
5489:
5487:
5485:Other results
5483:
5477:
5474:
5472:
5469:
5467:
5464:
5463:
5461:
5457:
5451:
5448:
5446:
5443:
5441:
5437:
5436:Hilbert space
5434:
5432:
5428:
5427:Inner product
5425:
5423:
5420:
5419:
5417:
5413:
5409:
5402:
5397:
5395:
5390:
5388:
5383:
5382:
5379:
5371:
5367:
5366:
5361:
5357:
5356:
5347:
5341:
5337:
5332:
5329:
5327:0-387-90227-9
5323:
5319:
5314:
5311:
5307:
5303:
5301:3-540-61786-8
5297:
5293:
5289:
5284:
5283:
5276:
5275:
5264:, p. 164
5263:
5262:Jacobson 2009
5258:
5251:
5247:
5240:
5231:
5224:
5218:
5214:
5207:
5200:
5196:
5192:
5188:
5184:
5180:
5173:
5160:
5157:
5150:
5143:
5138:
5136:
5129:
5124:
5120:
5113:
5107:
5103:
5098:
5092:
5087:
5083:
5076:
5070:
5068:
5067:Basic Algebra
5060:
5056:
5046:
5043:
5042:
5036:
5033:
5029:
5025:
5021:
5015:
5011:
5007:
5003:
4997:
4991:
4985:
4979:
4973:
4967:
4963:
4959:
4949:
4944:
4943:opposite ring
4939:
4933:
4929:
4925:
4920:
4916:
4912:
4906:
4903:
4897:
4891:
4885:
4879:
4872:
4866:
4860:
4856:
4849:
4845:
4841:
4837:
4833:
4829:
4823:
4817:
4815:
4810:
4804:
4798:
4794:
4768:
4765:
4762:
4756:
4750:
4747:
4741:
4738:
4735:
4729:
4722:
4721:
4720:
4717:
4712:
4707:
4703:
4699:
4695:
4689:
4686:
4680:
4676:
4669:
4665:
4661:
4654:
4650:
4646:
4641:
4637:
4632:
4628:
4624:
4620:
4614:
4611:
4607:
4601:
4597:
4590:
4586:
4582:
4576:
4572:
4566:
4560:
4555:
4550:
4544:
4541:
4535:
4531:
4525:
4519:
4515:
4511:
4507:
4484:
4478:
4471:
4468:
4465:
4459:
4455:
4452:
4446:
4443:
4440:
4437:
4434:
4428:
4421:
4404:
4401:
4398:
4392:
4389:
4383:
4380:
4377:
4371:
4368:
4362:
4359:
4356:
4350:
4347:
4341:
4338:
4335:
4329:
4326:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4296:
4289:
4288:
4287:
4285:
4284:-sesquilinear
4282:
4275:
4271:
4267:
4263:
4257:
4254:
4249:
4244:
4239:
4234:
4228:
4223:
4218:
4212:
4203:
4201:
4198:
4192:
4191:Reinhold Baer
4188:
4187:division ring
4183:
4179:
4173:
4167:
4160:
4156:
4152:
4147:
4146:nondegenerate
4142:
4137:
4133:
4124:
4118:
4112:
4106:
4102:
4098:
4094:
4090:
4089:
4088:
4085:
4081:
4076:
4072:
4064:
4060:
4056:
4053:
4052:
4046:
4044:
4038:
4034:
4028:
4022:
4016:
4011:
4006:
4002:
3998:
3978:
3973:
3964:
3960:
3954:
3950:
3946:
3941:
3932:
3928:
3922:
3918:
3914:
3909:
3900:
3896:
3890:
3886:
3882:
3876:
3873:
3870:
3864:
3857:
3856:
3855:
3852:
3842:
3835:
3828:
3824:
3814:
3807:
3800:
3796:
3791:
3786:
3779:
3775:
3771:
3766:
3755:
3753:
3747:
3742:
3736:
3729:
3723:
3717:
3710:
3705:
3700:
3694:
3691:
3684:
3680:
3673:
3666:
3662:
3655:
3652:
3647:
3643:
3638:
3634:
3630:
3626:
3621:
3615:
3611:
3607:
3587:
3582:
3579:
3575:
3571:
3568:
3565:
3556:
3550:
3544:
3537:
3521:
3518:
3514:
3510:
3504:
3498:
3491:
3490:
3489:
3486:
3480:
3473:
3469:
3462:
3460:
3456:
3451:
3446:
3441:
3434:
3429:
3424:
3417:
3397:
3394:
3384:
3381:
3378:
3372:
3366:
3363:
3357:
3354:
3351:
3345:
3338:
3337:
3336:
3333:
3327:
3323:
3317:
3311:
3306:
3302:
3298:
3290:
3284:
3273:
3256:
3253:
3247:
3244:
3241:
3235:
3215:
3212:
3206:
3203:
3200:
3194:
3187:
3186:
3185:
3182:
3176:
3172:
3167:
3162:
3151:
3149:
3148:
3142:
3138:
3132:
3128:
3123:
3119:
3113:
3109:
3105:
3099:
3093:
3089:
3083:
3079:
3074:
3068:
3062:
3058:
3053:
3048:
3044:
3024:
3018:
3015:
3001:
2998:
2995:
2984:
2973:
2970:
2967:
2964:
2953:
2948:
2944:
2936:
2935:
2934:
2931:
2925:
2920:
2915:
2909:
2904:
2899:
2893:
2885:Orthogonality
2882:
2879:
2873:
2867:
2847:
2844:
2837:
2834:
2831:
2825:
2818:
2812:
2809:
2803:
2800:
2797:
2794:
2791:
2785:
2778:
2777:
2776:
2773:
2767:
2763:
2757:
2751:
2747:
2741:
2737:
2736:division ring
2732:
2728:
2723:
2719:
2715:
2711:
2707:
2702:
2696:
2692:over a right
2691:
2688:
2676:
2674:
2669:
2658:
2656:
2637:
2634:
2631:
2625:
2622:
2617:
2607:
2593:
2591:
2586:
2570:
2567:
2562:
2559:
2552:
2536:
2524:
2521:
2518:
2512:
2506:
2503:
2497:
2494:
2491:
2485:
2457:
2454:
2451:
2448:
2445:
2437:
2433:
2423:
2410:
2407:
2404:
2401:
2393:
2389:
2385:
2366:
2363:
2360:
2354:
2351:
2346:
2336:
2322:
2320:
2315:
2313:
2294:
2291:
2288:
2276:
2274:
2256:
2252:
2248:
2245:
2240:
2236:
2232:
2223:
2221:
2220:Hilbert space
2217:
2216:inner product
2201:
2196:
2192:
2186:
2177:
2169:
2164:
2161:
2158:
2154:
2150:
2144:
2141:
2138:
2113:
2086:
2074:
2071:
2068:
2062:
2056:
2050:
2047:
2044:
2038:
2010:
2007:
2004:
2001:
1998:
1990:
1986:
1979:
1977:
1973:
1969:
1964:
1963:
1957:
1944:
1940:
1934:
1930:
1926:
1921:
1917:
1912:
1908:
1905:
1900:
1897:
1893:
1885:are given by
1872:
1864:
1846:
1842:
1821:
1818:
1815:
1810:
1806:
1802:
1798:
1792:
1788:
1784:
1779:
1775:
1770:
1766:
1761:
1757:
1746:
1742:
1734:
1730:
1724:
1720:
1716:
1712:
1706:
1702:
1696:
1692:
1686:
1682:
1678:
1673:
1669:
1663:
1659:
1653:
1649:
1644:
1640:
1637:
1631:
1628:
1625:
1619:
1612:and given by
1599:
1596:
1589:
1573:
1570:
1548:
1543:
1538:
1534:
1530:
1521:
1505:
1491:
1487:
1473:
1463:
1449:
1429:
1409:
1389:
1377:
1374:
1371:
1365:
1359:
1353:
1350:
1347:
1341:
1333:
1317:
1297:
1277:
1268:
1255:
1252:
1244:
1241:
1222:
1219:
1216:
1210:
1204:
1182:
1178:
1170:
1154:
1146:
1127:
1124:
1121:
1115:
1109:
1089:
1086:
1083:
1074:
1061:
1050:
1047:
1039:
1029:
1025:
1009:
1006:
998:
977:
947:
944:
936:
927:
922:
909:
906:
881:
859:
851:
848:
845:
842:
822:
819:
816:
813:
810:
807:
804:
801:
798:
768:
765:
762:
756:
752:
744:
739:
733:
730:
727:
724:
721:
715:
704:
701:
698:
692:
689:
683:
680:
677:
671:
668:
662:
659:
656:
650:
647:
641:
638:
635:
629:
626:
620:
617:
614:
611:
608:
605:
602:
596:
584:
583:
582:
560:
557:
554:
551:
548:
528:
521:
513:
509:
505:
502:
501:
499:
495:
485:
482:
468:
463:
459:
436:
426:
423:
420:
417:
409:
405:
402:
391:
389:
385:
381:
365:
356:
339:
331:
327:
318:
300:
291:
265:
260:
256:
250:
241:
233:
228:
225:
222:
218:
214:
208:
205:
202:
192:
191:
190:
187:
182:
181:Hilbert space
179:on a complex
178:
177:inner product
174:
170:
166:
156:
153:
149:
144:
139:
136:
129:
124:
123:division ring
120:
115:
113:
109:
105:
101:
96:
92:
88:
82:
77:
72:
70:
66:
65:
61:
57:
53:
49:
45:
41:
40:bilinear form
37:
33:
19:
5600:
5591:
5587:
5583:
5579:
5552:
5548:Self-adjoint
5459:Main results
5363:
5335:
5317:
5281:
5257:
5245:
5239:
5230:
5212:
5206:
5182:
5178:
5172:
5158:
5149:
5144:, p. 42
5125:, p. 28
5118:
5112:
5097:
5081:
5075:
5066:
5059:
5031:
5027:
5023:
5019:
5013:
5009:
5005:
5001:
4995:
4989:
4983:
4977:
4971:
4965:
4961:
4957:
4947:
4937:
4931:
4927:
4918:
4914:
4910:
4907:
4901:
4895:
4889:
4883:
4877:
4870:
4864:
4858:
4854:
4847:
4843:
4839:
4835:
4831:
4827:
4821:
4818:
4808:
4802:
4796:
4792:
4789:
4715:
4710:
4705:
4701:
4697:
4693:
4690:
4684:
4678:
4674:
4667:
4663:
4659:
4652:
4648:
4644:
4639:
4635:
4630:
4626:
4622:
4618:
4615:
4609:
4605:
4599:
4595:
4588:
4584:
4580:
4574:
4570:
4564:
4558:
4553:
4548:
4545:
4539:
4533:
4529:
4523:
4517:
4513:
4509:
4505:
4502:
4280:
4278:
4273:
4269:
4265:
4261:
4258:
4252:
4242:
4232:
4226:
4216:
4213:
4209:
4196:
4184:
4177:
4171:
4165:
4158:
4154:
4150:
4145:
4140:
4136:desarguesian
4130:is called a
4129:
4122:
4116:
4110:
4104:
4100:
4096:
4092:
4083:
4074:
4068:
4054:
4036:
4032:
4026:
4020:
4014:
4004:
4000:
3996:
3993:
3850:
3840:
3833:
3826:
3822:
3812:
3805:
3798:
3794:
3784:
3777:
3773:
3770:finite field
3764:
3761:
3751:
3745:
3740:
3734:
3732:. Then for
3727:
3721:
3715:
3708:
3704:identity map
3698:
3695:
3689:
3682:
3678:
3671:
3664:
3660:
3656:
3650:
3636:
3632:
3628:
3624:
3613:
3609:
3605:
3602:
3484:
3478:
3471:
3467:
3463:
3458:
3454:
3449:
3444:
3439:
3432:
3427:
3422:
3415:
3412:
3331:
3325:
3321:
3315:
3309:
3300:
3296:
3293:
3288:
3282:
3279:
3271:
3180:
3174:
3170:
3168:if, for all
3165:
3160:
3157:
3145:
3140:
3136:
3130:
3126:
3120:need not be
3111:
3107:
3103:
3097:
3091:
3087:
3081:
3077:
3072:
3066:
3060:
3056:
3051:
3046:
3042:
3039:
2929:
2923:
2918:
2913:
2907:
2897:
2891:
2888:
2877:
2871:
2865:
2862:
2771:
2765:
2761:
2755:
2749:
2745:
2739:
2730:
2721:
2717:
2713:
2709:
2700:
2694:
2686:
2684:
2682:
2667:
2664:
2594:
2587:
2435:
2431:
2429:
2382:is always a
2323:
2316:
2311:
2310:is called a
2277:
2224:
1988:
1984:
1982:
1967:
1965:
1497:
1402:In general,
1269:
1076:For a fixed
1075:
926:bilinear map
923:
790:
517:
503:
483:
397:
386:carrying an
379:
354:
315:denotes the
280:
189:is given by
185:
164:
162:
151:
142:
134:
127:
116:
94:
90:
86:
80:
73:
63:
35:
29:
5558:Trace class
4924:isomorphism
4546:An element
4132:correlation
4080:permutation
3790:prime power
3622:of the map
3620:fixed point
3154:Reflexivity
3085:(or simply
3040:Similarly,
2673:commutative
2384:real number
498:Dual system
44:dot product
32:mathematics
5617:Categories
5272:References
4814:involution
4719:such that
4554:orthogonal
4059:antilinear
4055:Assumption
4024:is then a
4018:. The map
4010:involutory
3750:is called
3743:, and for
3305:-Hermitian
3292:is called
3070:, written
3052:orthogonal
2679:Definition
2478:such that
2430:A complex
2031:such that
1983:A complex
1243:functional
1169:dual space
508:antilinear
504:Assumption
492:See also:
394:Convention
104:antilinear
56:semilinear
5370:EMS Press
5215:, Dover,
5086:D. Reidel
5022:′ :
4993:-module,
4757:φ
4751:σ
4730:φ
4711:Hermitian
4636:reflexive
4568:(written
4479:σ
4460:φ
4429:φ
4393:φ
4372:φ
4351:φ
4330:φ
4297:φ
3865:φ
3741:symmetric
3580:−
3576:ε
3572:α
3569:ε
3557:α
3551:σ
3545:σ
3519:−
3515:ε
3505:ε
3499:σ
3455:Hermitian
3447:. (When
3430:, and if
3428:Hermitian
3395:ε
3373:φ
3367:σ
3346:φ
3236:φ
3195:φ
3166:reflexive
3144:(but see
3122:symmetric
3016:∈
3008:∀
2974:φ
2971:∣
2965:∈
2949:⊥
2903:submodule
2845:β
2826:φ
2819:α
2813:σ
2804:β
2795:α
2786:φ
2568:−
2532:¯
2507:−
2461:→
2455:×
2405:∈
2257:∗
2246:−
2241:∗
2181:¯
2155:∑
2148:⟩
2136:⟨
2082:¯
2014:→
2008:×
1966:The term
1909:φ
1847:†
1811:†
1767:φ
1752:¯
1731:∑
1721:∑
1683:∑
1650:∑
1641:φ
1620:φ
1474:φ
1464:Hermitian
1450:φ
1430:φ
1410:ψ
1385:¯
1366:φ
1342:ψ
1318:ψ
1278:φ
1211:φ
1208:↦
1183:∗
1116:φ
1113:↦
1087:∈
1054:→
1048:⊗
1043:¯
981:¯
951:→
945:×
940:¯
885:¯
852:∈
820:∈
757:φ
748:¯
716:φ
693:φ
672:φ
651:φ
630:φ
597:φ
564:→
558:×
549:φ
464:∗
427:∈
295:¯
245:¯
219:∑
212:⟩
200:⟨
175:, as the
5572:Examples
5039:See also
5004: :
4981:-module
4935:, where
4913: :
4852:for all
4825:we have
4790:for all
4696: :
4672:for all
4657:implies
4621: :
4527:and all
4503:for all
4264: :
4200:-modules
4163:for all
3999: :
3994:The map
3782:, where
3642:subgroup
3228:implies
3150:below).
3118:relation
3116:. This
2759:and all
2712: :
2698:-module
1330:via the
1102:the map
835:and all
791:for all
5586:) with
5563:Unitary
5422:Adjoint
5372:, 2001
5310:0233275
5199:1968621
5104:at the
4941:is the
4875:, then
4061:(resp.
3758:Example
3706:(i.e.,
3702:is the
3644:of the
3124:, i.e.
2273:SU(1,1)
1861:is the
1022:By the
995:is the
518:Over a
401:Dirac's
173:physics
138:-module
64:sesqui-
5543:Normal
5342:
5324:
5308:
5298:
5219:
5197:
5165:1 = −1
5045:*-ring
4812:is an
4259:A map
4238:module
4063:linear
4008:is an
3005:
2917:, the
1834:where
1588:matrix
968:where
872:Here,
541:a map
512:linear
337:
281:where
69:scalar
52:linear
5594:<∞
5195:JSTOR
5153:When
5051:Notes
4868:, if
4670:) = 0
4655:) = 0
4642:) if
4591:) = 0
4578:) if
4220:be a
4161:) = 0
4069:In a
3788:is a
3776:= GF(
3618:is a
3114:) = 0
2734:of a
2704:is a
1974:on a
1520:basis
1238:is a
1143:is a
384:rings
148:rings
108:field
5516:Maps
5438:and
5429:and
5340:ISBN
5322:ISBN
5296:ISBN
5217:ISBN
5156:char
4873:= id
4834:) =
4638:(or
4240:and
4222:ring
4214:Let
4078:, a
3854:by:
3820:and
3762:Let
3748:= −1
3711:= id
3435:= −1
1422:and
496:and
34:, a
5187:doi
5161:= 2
4945:of
4862:in
4800:in
4709:is
4682:in
4634:is
4552:is
4537:in
4521:in
4286:if
4277:is
4250:of
4246:an
4230:an
4180:= 0
4169:in
4148:if
4144:is
4120:of
3825:= (
3797:= (
3737:= 1
3730:= 1
3713:),
3648:of
3482:in
3461:.)
3457:or
3418:= 1
3413:If
3329:in
3313:in
3178:in
3164:is
3095:if
3054:to
3050:is
2933:is
2921:of
2911:of
2769:in
2753:in
2671:is
1563:of
1498:If
1290:on
1245:on
1147:on
1026:of
999:to
451:as
406:in
319:of
167:on
46:of
30:In
5619::
5368:,
5362:,
5306:MR
5304:,
5294:,
5286:,
5248:,
5193:,
5183:37
5181:,
5134:^
5127:–
5121:,
5090:–
5084:,
5035:.
5030:→
5026:×
5012:→
5008:×
4966:ba
4964:=
4960:∗
4930:→
4917:→
4857:,
4832:st
4795:,
4704:→
4700:×
4688:.
4677:,
4666:,
4651:,
4629:→
4625:×
4613:.
4608:⊥
4598:⊥
4587:,
4573:⊥
4543:.
4532:,
4516:,
4512:,
4508:,
4272:→
4268:×
4256:.
4224:,
4189:,
4182:.
4157:,
4114:,
4103:⊆
4099:⇒
4095:⊆
4003:↦
3839:,
3832:,
3811:,
3804:,
3754:.
3693:.
3681:,
3663:,
3657:A
3654:.
3627:↦
3612:,
3488:,
3470:,
3335:,
3324:,
3299:,
3280:A
3257:0.
3184:,
3173:,
3139:⊥
3129:⊥
3110:,
3090:⊥
3059:∈
3045:∈
2905:)
2881:.
2775:,
2764:,
2748:,
2720:→
2716:×
2683:A
2657:.
2592:.
2563::=
2321:.
2314:.
2275:.
1906::=
1334::
481:.
155:.
114:.
93:→
89:×
78:,
5601:F
5592:n
5588:K
5584:K
5582:(
5580:C
5508:)
5504:(
5400:e
5393:t
5386:v
5189::
5159:K
5032:R
5028:V
5024:V
5020:φ
5014:R
5010:V
5006:V
5002:φ
4996:V
4990:R
4984:V
4978:R
4972:R
4962:b
4958:a
4953:∗
4948:R
4938:R
4932:R
4928:R
4919:R
4915:R
4911:σ
4902:V
4896:R
4890:R
4884:φ
4878:R
4871:σ
4865:R
4859:t
4855:s
4850:)
4848:s
4846:(
4844:σ
4842:)
4840:t
4838:(
4836:σ
4830:(
4828:σ
4822:σ
4809:σ
4803:V
4797:y
4793:x
4775:)
4772:)
4769:x
4766:,
4763:y
4760:(
4754:(
4748:=
4745:)
4742:y
4739:,
4736:x
4733:(
4716:σ
4706:R
4702:V
4698:V
4694:φ
4685:V
4679:y
4675:x
4668:x
4664:y
4662:(
4660:φ
4653:y
4649:x
4647:(
4645:φ
4631:R
4627:V
4623:V
4619:φ
4610:x
4606:y
4600:y
4596:x
4589:y
4585:x
4583:(
4581:φ
4575:y
4571:x
4565:φ
4559:y
4549:x
4540:R
4534:d
4530:c
4524:V
4518:w
4514:z
4510:y
4506:x
4488:)
4485:d
4482:(
4475:)
4472:y
4469:,
4466:x
4463:(
4456:c
4453:=
4450:)
4447:y
4444:d
4441:,
4438:x
4435:c
4432:(
4408:)
4405:w
4402:,
4399:y
4396:(
4390:+
4387:)
4384:z
4381:,
4378:y
4375:(
4369:+
4366:)
4363:w
4360:,
4357:x
4354:(
4348:+
4345:)
4342:z
4339:,
4336:x
4333:(
4327:=
4324:)
4321:w
4318:+
4315:z
4312:,
4309:y
4306:+
4303:x
4300:(
4281:σ
4274:R
4270:V
4266:V
4262:φ
4253:R
4243:σ
4236:-
4233:R
4227:V
4217:R
4197:R
4178:x
4172:V
4166:y
4159:y
4155:x
4153:(
4151:φ
4141:φ
4126:,
4123:G
4117:T
4111:S
4105:S
4101:T
4097:T
4093:S
4084:δ
4075:G
4037:φ
4033:M
4027:σ
4021:φ
4015:F
4005:t
4001:t
3997:σ
3979:.
3974:q
3965:3
3961:y
3955:3
3951:x
3947:+
3942:q
3933:2
3929:y
3923:2
3919:x
3915:+
3910:q
3901:1
3897:y
3891:1
3887:x
3883:=
3880:)
3877:y
3874:,
3871:x
3868:(
3851:φ
3846:)
3844:3
3841:y
3837:2
3834:y
3830:1
3827:y
3823:y
3818:)
3816:3
3813:x
3809:2
3806:x
3802:1
3799:x
3795:x
3785:q
3780:)
3778:q
3774:F
3765:V
3746:ε
3735:ε
3728:ε
3722:φ
3716:K
3709:σ
3699:σ
3690:ε
3685:)
3683:ε
3679:σ
3677:(
3672:σ
3667:)
3665:ε
3661:σ
3659:(
3651:K
3637:ε
3635:)
3633:α
3631:(
3629:σ
3625:α
3616:)
3614:x
3610:x
3608:(
3606:φ
3588:.
3583:1
3566:=
3563:)
3560:)
3554:(
3548:(
3522:1
3511:=
3508:)
3502:(
3485:K
3479:α
3474:)
3472:ε
3468:σ
3466:(
3450:σ
3443:-
3440:σ
3433:ε
3426:-
3423:σ
3416:ε
3398:.
3391:)
3388:)
3385:x
3382:,
3379:y
3376:(
3370:(
3364:=
3361:)
3358:y
3355:,
3352:x
3349:(
3332:M
3326:y
3322:x
3316:K
3310:ε
3303:)
3301:ε
3297:σ
3295:(
3289:φ
3283:σ
3254:=
3251:)
3248:x
3245:,
3242:y
3239:(
3216:0
3213:=
3210:)
3207:y
3204:,
3201:x
3198:(
3181:M
3175:y
3171:x
3161:φ
3141:x
3137:y
3131:y
3127:x
3112:y
3108:x
3106:(
3104:φ
3098:φ
3092:y
3088:x
3082:y
3078:φ
3075:⊥
3073:x
3067:φ
3061:M
3057:y
3047:M
3043:x
3025:.
3022:}
3019:W
3012:w
3002:,
2999:0
2996:=
2993:)
2989:w
2985:,
2981:v
2977:(
2968:M
2961:v
2957:{
2954:=
2945:W
2930:φ
2924:W
2914:M
2908:W
2898:M
2892:φ
2878:φ
2872:φ
2866:σ
2848:.
2841:)
2838:y
2835:,
2832:x
2829:(
2822:)
2816:(
2810:=
2807:)
2801:y
2798:,
2792:x
2789:(
2772:K
2766:β
2762:α
2756:M
2750:y
2746:x
2740:K
2731:σ
2722:K
2718:M
2714:M
2710:φ
2701:M
2695:K
2687:σ
2668:K
2641:)
2638:z
2635:,
2632:z
2629:(
2626:s
2623:=
2618:s
2613:|
2608:z
2604:|
2571:1
2560:i
2537:.
2528:)
2525:w
2522:,
2519:z
2516:(
2513:s
2504:=
2501:)
2498:z
2495:,
2492:w
2489:(
2486:s
2465:C
2458:V
2452:V
2449::
2446:s
2411:.
2408:V
2402:z
2370:)
2367:z
2364:,
2361:z
2358:(
2355:h
2352:=
2347:h
2342:|
2337:z
2333:|
2298:)
2295:h
2292:,
2289:V
2286:(
2253:z
2249:z
2237:w
2233:w
2202:.
2197:i
2193:z
2187:i
2178:w
2170:n
2165:1
2162:=
2159:i
2151:=
2145:z
2142:,
2139:w
2114:n
2109:C
2087:.
2078:)
2075:w
2072:,
2069:z
2066:(
2063:h
2057:=
2054:)
2051:z
2048:,
2045:w
2042:(
2039:h
2018:C
2011:V
2005:V
2002::
1999:h
1978:.
1945:.
1941:)
1935:j
1931:e
1927:,
1922:i
1918:e
1913:(
1901:j
1898:i
1894:A
1873:A
1843:w
1822:.
1819:z
1816:A
1807:w
1803:=
1799:)
1793:j
1789:e
1785:,
1780:i
1776:e
1771:(
1762:j
1758:z
1747:i
1743:w
1735:j
1725:i
1717:=
1713:)
1707:j
1703:e
1697:j
1693:z
1687:j
1679:,
1674:i
1670:e
1664:i
1660:w
1654:i
1645:(
1638:=
1635:)
1632:z
1629:,
1626:w
1623:(
1600:,
1597:A
1574:,
1571:V
1549:i
1544:}
1539:i
1535:e
1531:{
1506:V
1390:.
1381:)
1378:w
1375:,
1372:z
1369:(
1360:=
1357:)
1354:z
1351:,
1348:w
1345:(
1298:V
1256:.
1253:V
1226:)
1223:z
1220:,
1217:w
1214:(
1205:w
1179:V
1155:V
1131:)
1128:w
1125:,
1122:z
1119:(
1110:w
1090:V
1084:z
1062:.
1058:C
1051:V
1040:V
1010:.
1007:V
978:V
955:C
948:V
937:V
910:.
907:a
882:a
860:.
856:C
849:b
846:,
843:a
823:V
817:w
814:,
811:z
808:,
805:y
802:,
799:x
772:)
769:y
766:,
763:x
760:(
753:b
745:a
740:=
737:)
734:y
731:b
728:,
725:x
722:a
719:(
708:)
705:w
702:,
699:y
696:(
690:+
687:)
684:z
681:,
678:y
675:(
669:+
666:)
663:w
660:,
657:x
654:(
648:+
645:)
642:z
639:,
636:x
633:(
627:=
624:)
621:w
618:+
615:z
612:,
609:y
606:+
603:x
600:(
568:C
561:V
555:V
552::
529:V
469:z
460:w
437:n
432:C
424:z
421:,
418:w
366:i
355:C
340:.
332:i
328:w
301:i
292:w
266:.
261:i
257:z
251:i
242:w
234:n
229:1
226:=
223:i
215:=
209:z
206:,
203:w
186:C
152:R
143:R
135:K
128:K
95:C
91:V
87:V
81:V
20:)
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