1640:"The entire space is spanned by some orthonormal set of eigenvectors of A." I could not find any material stating this, but rather statements equivalent with the eigenvectors of A spanning the entire space. If so, the vectors are generating the space, thus a basis can be selected. If a basis can be selected, then via the Gram-Schmidt orthogonalization, an orthogonal set of vectors can be constructed, which are still the eigenvectors of the matrix, and normalizing them does not change that. Is this reasoning sound? If so, wouldn't it be better to elaborate a bit in the article?
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1601:"Normality is a convenient test for diagonalizability: every normal matrix can be converted to a diagonal matrix by a unitary transform, and every matrix which can be made diagonal by a unitary transform is also normal, but finding the desired transform requires much more work than simply testing to see whether the matrix is normal." --
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There seems to be a problem with equivalence number 10: "A commutes with some normal matrix N with distinct eigenvalues" if and only if A is normal. I think this is only true if A has N distinct eigenvalues. If A is normal with some degeneracy, any matrix commuting with A will share its eigenspaces,
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Ok... I think I'm with you. Just checked the proof I was looking at. If A commutes with B and B is normal, then every k-dimensional eigenspace of A has to contain an orthogonal set of k eigenvectors for B. I interpreted that as eigenspace of B, which it obviously doesn't have to be. B could have N
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I think the point is that a normal matrix is similar to a diagonal matrix with complex entries. If all of the entries are real/imag/unit/positive then the matrix itself is
Hermitian/AnitHermitian/Unitary/Positive. Since addition and multiplication of diagonal matrices act individually on each
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Thanks for spotting the error and sorry for the trouble. I got mixed up when I was writing the first paragraph... Is the set of normal matrices that are a subset of EP matrices. But looking back now and seeing the EP matrices article, I think this article is better without this contribution.
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The "analogy" section of this page states that it is "occasionally useful" to make the analogy with complex numbers - but it doesn't give any explanation of why or in what way it can be useful. Without such information the statement of the analogy doesn't seem very helpful.
269:(1) Such words are sometimes uncapitalized if they are being used intensively, by specialists in a particular area. But I don't think 'Hermitian' is usually in this category. In an encyclopaedia, I would leave the capital in. (Likewise with 'Laplacian', 'Lagrangian' etc.)
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Distinct eigenvalues means N has one dimensional eigenspaces corresponding to each eigenvalue. So the statement is OK. The fact that two normal matrices commute say nothing about their kernels. Take, for example, the identity matrix and the zero matrix.
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Normality is equivalent to unitary diagonalizability, which is stronger than diagonalizability. But to actually find the unitary transform that diagonalizes the matrix means to find the eigenvectors, which from what I hear can be numerically tricky.
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How widely is normality used as a test for diagonalizability? I personally found the second paragraph confusing since it seems to imply that many or most diagonalizable matrices are normal, and normality is a quick and easy way to test for it.
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Perhaps something like above could be added to this section. There are already a few similar
Propositions in the "Consequences" section, but they are separately stated and not tied together, which obscures the generality of the normal matrix
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Thanks. That's a good way of putting it. A is then diagonalisable in the unique basis that diagonalises B... but not necessarily uniquely. If A has any multiple eigenvalues then the diagonal form of A will contain blocks of
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Correction : a positive definite matrix is not normal ! Take a small 2 by 2 example to see that : $ $ A=\left(\begin{array}{cc}5&1\\0&7\end{array})\right.$ $ The matrix $ A$ is positive definite but non normal.
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1699:. Certainly a reader new to the subject might not see this immediately, but to me this point does not stand out as needing a citation among other statements of the same nature in the article. Tag removed.
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Orthonormality follows trivially from that unitary diagonalizability of a normal matrix. Having a set of eigenvectors which forms a basis (mere diagonalizability) is strictly weaker than normality.
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many terms in this stub are not defined. This article either needs expanding to explain the terms, to have the terms converted into links where they are explained or the be merged into the main
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There is an obvious misunderstanding of the terms "special case" and "sufficient". Since I am not an expert in this field, I decided to remove the contribution instead of rectifying it.
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entry, normal matrices share the same properties as the corresponding complex numbers. This only works for more multiple matrices if they are simultaneously diagonalizable, however.
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of being positive definite. A misjudgment, surely? - it certainly confused me for a while. Note to self or request to others - check
Knowledge's article on positive definiteness.
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is an even simpler example. I think the original author assumed positive definiteness implied symmetry. This isn't necessarily the case for real matrices. (See the article on
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is
Hermitian, but this is not true in general for real matrices. perhaps Horn and Johnson required Hermiticity in both the real and complex cases to get a uniform definition?
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I suggest getting rid of the fancy boxes. They contribute nothing to the article except a graphic design, which is unnecessary. (I find them intrusive.) Opinions?
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1991:👍 I haven't checked out the boxes on this page, but on other pages I've seen colored boxes cause the equations to be illegible when viewed in dark mode on mobile.
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272:(2) Positive definite. Horn and Johnson,'Matrix Analysis', p396, make being Hermitian (and therefore symmetric, in the case of real matrices) part of the
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Should "Hermitian" be written with a capital 'H'? I think it always is everywhere else I've seen it, but I don't know what accepted style is for this. --
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Can anybody link a proof (or references) of the statement "the columns of U* are eigenvectors of both A and B"? (last phrase of section "consequences")
711:{\displaystyle B={\begin{pmatrix}\lambda _{1}I_{k_{1}}&0&0\\0&\lambda _{2}I_{k_{2}}&0\\0&0&\lambda _{3}I_{k_{3}}\end{pmatrix}}}
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304:. Anyway, the latter article does not say clearly whether Hermitian is required, precisely because the literature is confused about this issue.
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is a rank-1 projection. for a similar finite dimensional example, consider the square matrix that is 1 on the superdiagonal and 0 elsewhere.
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I didn't add that section, but after reading it I found it useful. If nothing else, it helps understand what the defintions mean. :)
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1402:{\displaystyle \sum _{j=1}^{n}A_{ij}(U^{*})_{jk}=\sum _{j=1}^{n}(U^{*})_{ij}{\Lambda }_{jk}=(U^{*})_{ik}{\lambda }_{k}}
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are also a special case of normal matrices. EPness, however, is not a sufficient condition to normality. For example,
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denotes the k by k identity matrix. Now, in the same basis, a matrix A commutes with B if and only if A is of the form
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898:{\displaystyle A={\begin{pmatrix}M_{k_{1}}&0&0\\0&M_{k_{2}}&0\\0&0&M_{k_{3}}\end{pmatrix}}}
307:(3) I tend to agree, but I'm not totally sure what you mean and I don't fully grasp the concept of "normal matrix".
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In the classification, Dirac and Pauli matrices should be in the same category, since they are so similar.--
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be the matrix having the basis vectors as columns; it is unitary, by orthonormality. In the product
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distinct eigenvalues and still satisfy that condition. Thanks. (Let me know if I'm not getting it)
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Knowledge. If you would like to participate, please visit the project page, where you can join
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but among 2×2 matrices, there are only ones that are multiples of unitary matrices.
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Let's look at it this way. Take a normal matrix B. If B has, say, 3 eigenvalues λ
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normal would aid understanding. NTSORTO - insert some discussion accordingly.
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I have doubts about the last sentence of comment in source in #Special cases
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and therefore share its degeneracy. Can anyone check this for me please?
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My matrix stuff was too many years ago for me to be able to fix this. --
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I have undone changes, which introduced the following text:
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respectively. So in some orthonormal basis, B takes the form
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is an example of a non-normal operator. the commutator
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1002:{\displaystyle \lambda I_{k}}
109:and see a list of open tasks.
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1986:10:36, 3 December 2023 (UTC)
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1566:{\displaystyle \mathbf {B} }
1097:{\displaystyle \mathbf {U} }
1075:{\displaystyle \mathbf {A} }
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298:Talk:Cholesky decomposition
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1593:Test for diagonalizability
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1695:. Multiply both sides by
1551:You can do the same with
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765:
764:
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748:
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727:
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705:
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510:
492:
491:
459:
456:
436:
435:chicken vs egg
433:
422:Chris2crawford
407:Chris2crawford
402:
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370:
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344:
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176:Capitalization
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107:the discussion
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26:
13:
10:
9:
6:
4:
3:
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2033:
2022:
2019:
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2011:
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1998:
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1983:
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1845:
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1840:Saung Tadashi
1836:
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1507:
1444:
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1413:
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1379:
1369:
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1234:
1196:
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1177:
1173:
1169:
1164:
1161:
1129:
1059:
1058:
1057:
1054:
1052:
1048:
1044:
1043:77.168.20.183
1040:
1030:
1020:
1016:
1012:
994:
990:
986:
977:
976:
975:
974:
973:
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933:
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871:
865:
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844:
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835:
829:
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792:
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784:
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771:
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768:
746:
742:
733:
732:
731:
730:
729:
728:
703:
693:
689:
684:
678:
674:
668:
663:
656:
647:
643:
638:
632:
628:
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615:
610:
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586:
582:
575:
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567:
560:
559:
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519:
518:
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509:
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501:
496:
495:
494:
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490:
486:
482:
477:
476:
475:
474:
470:
466:
457:
455:
454:
450:
446:
442:
441:Matrix Theory
434:
432:
431:
427:
423:
417:
416:
412:
408:
400:
396:
392:
388:
387:
386:
385:
381:
377:
368:
364:
361:
357:
353:
350:
346:
345:
341:
338:implies that
337:
332:
328:
324:
323:
320:
316:
312:
309:
306:
303:
299:
295:
292:
291:
290:
289:7 March 2007
288:
284:
282:
277:
275:
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206:
196:
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187:
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183:
175:
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164:
160:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1975:
1954:
1884:
1876:
1857:
1821:
1817:
1722:
1718:
1696:
1692:
1688:
1684:
1680:
1676:
1672:
1639:
1631:
1600:
1596:
1055:
1037:— Preceding
1034:
461:
458:Equivalences
440:
438:
418:
403:
372:
355:
351:
339:
335:
334:≥ 0 for all
330:
326:
311:Jitse Niesen
293:(1) I agree.
285:
280:
278:
273:
271:
268:
191:
179:
167:
158:
156:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
1972:Fancy boxes
1885:The matrix
1724:EP matrices
197:The matrix
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2010:Categories
1701:Styrofoams
1579:Bobathon71
1011:Bobathon71
500:Bobathon71
465:Bobathon71
391:Shreevatsa
274:definition
356:T*T - TT*
182:DudeGalea
1642:Andorxor
1471:, where
1039:unsigned
420:analogy.
347:(3) the
170:SGBailey
1959:Hbghlyj
1657:Mct mht
1619:Mct mht
1603:Aluchko
1500:is the
1104:, then
954:Mct mht
540:, and k
528:, and λ
481:Mct mht
360:Mct mht
139:on the
1978:Zaslav
921:where
734:where
445:Lavaka
287:Rwb001
165:page.
163:matrix
36:scale.
333:: -->
256:Lunch
1997:talk
1982:talk
1963:talk
1864:talk
1844:talk
1828:talk
1705:talk
1661:talk
1646:talk
1623:talk
1607:talk
1583:talk
1047:talk
1015:talk
958:talk
504:talk
485:talk
469:talk
449:talk
426:talk
411:talk
395:talk
380:talk
315:talk
254:.)
1060:If
536:, k
524:, λ
281:not
159:Too
131:Mid
2012::
1999:)
1984:)
1965:)
1912:−
1866:)
1846:)
1830:)
1707:)
1693:UΛ
1691:=
1689:AU
1685:AU
1663:)
1648:)
1625:)
1609:)
1585:)
1534:∗
1441:λ
1390:λ
1370:∗
1346:Λ
1326:∗
1298:∑
1276:∗
1235:∑
1211:∗
1174:λ
1157:Λ
1134:Λ
1124:∗
1049:)
1017:)
987:λ
960:)
675:λ
629:λ
583:λ
506:)
487:)
471:)
451:)
428:)
413:)
397:)
382:)
329:,
327:Ax
317:)
1995:(
1980:(
1961:(
1955:A
1939:]
1933:i
1930:+
1927:1
1922:1
1915:1
1907:1
1901:[
1896:=
1893:A
1862:(
1842:(
1826:(
1801:)
1795:0
1790:0
1785:0
1778:0
1773:4
1768:3
1761:0
1756:2
1751:1
1745:(
1740:=
1737:A
1703:(
1697:U
1681:U
1677:U
1673:U
1659:(
1644:(
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1581:(
1573:.
1560:B
1530:U
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1485:k
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1456:k
1452:u
1445:k
1437:=
1431:k
1427:u
1423:A
1409:.
1395:k
1383:k
1380:i
1376:)
1366:U
1362:(
1359:=
1354:k
1351:j
1339:j
1336:i
1332:)
1322:U
1318:(
1313:n
1308:1
1305:=
1302:j
1294:=
1289:k
1286:j
1282:)
1272:U
1268:(
1263:j
1260:i
1256:A
1250:n
1245:1
1242:=
1239:j
1207:U
1192:.
1178:i
1170:=
1165:i
1162:i
1130:=
1120:U
1116:A
1113:U
1091:U
1069:A
1045:(
1013:(
995:k
991:I
956:(
950:i
934:k
930:M
891:)
881:3
877:k
872:M
866:0
861:0
854:0
845:2
841:k
836:M
830:0
823:0
818:0
809:1
805:k
800:M
793:(
788:=
785:A
747:k
743:I
704:)
694:3
690:k
685:I
679:3
669:0
664:0
657:0
648:2
644:k
639:I
633:2
623:0
616:0
611:0
602:1
598:k
593:I
587:1
576:(
571:=
568:B
542:3
538:2
534:1
530:3
526:2
522:1
502:(
483:(
467:(
447:(
424:(
409:(
393:(
378:(
352:T
340:A
336:x
331:x
313:(
236:]
230:1
225:0
218:1
213:1
207:[
143:.
42::
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