Talk:Hermitian matrix

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Hermitian matrices are defined much more generally. This article only deals with the most common case, complex numbers. You just need and a field and a fixed involutive automorphism and you can define hermitian matrix. It is often done over finite fields, and it can also be define with other involutive automorphisms of the complex numbers than the usual conjugation.Evilbu 22:02, 4 February 2006 (UTC)

1)I've heard only this about automorphisms of the complex field besides the triv. and conj: that they're everywhere discontinuous, & there's a lot of them. Is it actully known whether any (or all)besides conj. are of order two? 2)Has anyone worked on what to do for the generalization of symmetric matrices to "hermitian" if one has, say, a field with an automorphism group of order 3, or worse, an automorphism group isomorphic to the symmetric group on three elements? I'm very interested.Rich 08:18, 6 October 2006 (UTC) ,

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[edit] Conjugate transpose

A* is the conjugate matrix of A. Atranspose is the transposed matric of A. The conjugate transpose is denoted A*transpose. I don't know off hand how to do the transpose symbol in Latex. But it would look roughly like A*t.

A* is the standard notation for conjugate transpose in matrix theory, but may well denote the conjugate matrix in other displines further removed from maths. See Talk:Conjugate matrix for some previous discussion. -- Jitse Niesen (talk) 04:24, 10 June 2006 (UTC)

For clarity, we can use AH to denote the matrix conjugate transpose. (Aug.12th, 2006)

[edit] anti commutative matrices can't be hermitian??

I think AB=BA must not be true for hermitian matrix. For example think of a1:matrix([0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]) and a2:matrix([0,0,0,-%i],[0,0,%i,0],[0,-%i,0,0],[%i,0,0,0]). a1 * a2 = matrix([%i,0,0,0],[0,-%i,0,0],[0,0,%i,0],[0,0,0,-%i])!=matrix([-%i,0,0,0],[0,%i,0,0],[0,0,-%i,0],[0,0,0,%i])=a2 * a1.

Or am I wrong?? (the matrices i took above are the dirac matrices)

All diagonal matrices are Hermitian and commute, so AB = BA holds for that family. I hope that answers your question. -- Jitse Niesen (talk) 16:49, 8 January 2007 (UTC)
slight clarification: All diagonal matrices with real entries are Hermitian and commute. Mct mht 17:00, 8 January 2007 (UTC)

[edit] Delete the commas in the first equation

How 'bout deleting the commas between the i,j subscripts?

[edit] some more intuitive explanations please?

I personally find intuitive explanations very important when learning math. I have written an article: [1] "Visualizing Hermitian Matrix as An Ellipse with Dr. Geo" for my students and wonder if it is appropriate to link to it from here. —Preceding unsigned comment added by Ckhung (talk • contribs) 02:11, August 28, 2007 (UTC)

That looks appropriate. I think your explanation is not succinct enough to include in an encyclopaedia but it does add to the article. So I added the link. Thanks. -- Jitse Niesen (talk) 03:44, 28 August 2007 (UTC)
that works only for positive matrix though. maybe relocate the link to that article? one particular instance is the inertia ellipsoid from mechanics. Mct mht 04:59, 28 August 2007 (UTC)