Talk:Hermitian matrix

Initial Comments

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)

Conjugate transpose

A* is the conjugate matrix of A. A^transpose 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 $A^{H}$ to denote the matrix conjugate transpose. (Aug.12th, 2006)

In the physics community and in many math books that I've seen, it's common to use a superscripted dagger to represent a Hermitian conjugate (as in A^{$\scriptstyle \dagger$}). To me, this seems like an ideal notation because it won't confuse anyone whereas the * notation will probably confuse all of the physicists and engineers and many of the math students who visit this page.

Also, I noticed that the dagger notation is in use further down the page, and I think we can all agree that the same notation should be used throughout the page. I'm going to change the first appearance to dagger. Milez (talk) 01:48, 22 November 2008 (UTC)

Would it not be better to use A^H instead? The dagger notation is often used as the pseudo-inverse, whereas either A* or A^H are more common. Mmcmcmc (talk) 15:08, 28 November 2008 (UTC)

I believe the definition of complex conjugation and discussion of Hemiticity as invariance under the adjunct only applies to Euclidian normed spaces. I am looking at a 2+2 vector space where the basis is decomposed into 2 vectors of positive norm and 2 vectors of negative norm and this no longer hold true. Since Hermiticity is well defined regardless of norm, it still makes sense in such a space, but the adjunct is defined as gamma_0 * A * gamma_0 where gamma_0 is pure diaganol with 1's and -1's pertaining to the norm of the associated dimension (positive or negative) — Preceding unsigned comment added by Jjrusnak (talk • contribs) 18:51, 7 February 2021 (UTC)

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)

Delete the commas in the first equation

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

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)

Section on "Circles"

The last section on circles looks like a very specific use of Hermitian matrices, rather than general information. Perhaps it should be moved into an "Examples" section with a few other bullet points, or just outright deleted? —Preceding unsigned comment added by Evanpw (talk • contribs) 22:37, 24 July 2008 (UTC)

General Corrections

I had to change the statement about the dimension of the vector space of real symmetric matrices from n^2 to n(n+1)/2. Someone had to have miscounted, because any n^2 dimensional subspace of M_n(R) is, of course, all of M_n(R). Also some claification was needed for some 10%-true statements like: "The eigenvectors of a Hermitian matrix are orthogonal". I also considered the statement " moreover, eigenvectors with distinct eigenvalues are orthogonal" to be irrelevant, since this isn't a special property of Hermetian matrices and it doesn't seem to be used later. The important information is that the geometric multiplicity of the eigenvalues matches the algebraic multiplicity. I hope the switch from \sigma to \lambda isn't too bad, I did that to come in line with the Eigendecomposition article, and to avoid confusion with the singular value decomposition.Rschwieb (talk) 20:14, 3 February 2011 (UTC)

Again, "dimension n^2" is incorrect, and should be replaced with n(n+1)/2. If you doubt the basis I gave, the fact of the matter is confirmed in an Algebra book and at Cliff's Notes. Rschwieb (talk) 17:26, 5 February 2011 (UTC)

UPDATE: Somehow at the time I had misread that "over the reals" as "matrices over the reals" and subsequently posted info for real Hermitian (=symmetric) matrices. I see now that it meant "complex matrices as vector spaces over reals". Sorry for the confusion. Now I see what the anonymous editor was getting at, however it would have been a lot more helpful if they had said something sooner here. Rschwieb (talk) 15:24, 23 March 2011 (UTC)

"with" to "that may have"

I changed "with" to "that may have" in the first sentence. This is because on the Physics Forum site

https://www.physicsforums.com/threads/neutrino-theory-regarding-rest-masses.820577/page-5#post-5157225

I quoted the previous form of the first sentence of this article with an interpretation that this excluded real matrices, and I was corrected. I had read just the first sentence to refresh my memory about the definition. I think the first sentence being a definition is a good idea, and that a reader should not be required to read an entire article to understand the definition correctly. BuzzBloom (talk) 14:14, 2 July 2015 (UTC)

Buzz, you have been told several times in-thread that this edit is not a good one. I have reverted your edits. Blennow (talk) 06:20, 3 July 2015 (UTC)

difference of a square matrix and its conjugate transpose

"The difference of a square matrix and its conjugate transpose (𝐴 − 𝐴^𝐻) is skew-Hermitian (also called antihermitian). This implies that the commutator of two Hermitian matrices is skew-Hermitian."

This doesn't make sense to me. I think this last sentence should be difference that, or thm that states that product of Hermitian matrices is Hermitian iff they commute is incorrect. Sorry for bad formatting! Ehaarer (talk) 20:22, 25 July 2023 (UTC)

Relation to symmetric matrices

I'd like an intuitive explanation of this, or what the author had in mind writing it: "Hermitian matrices can be understood as the complex extension of real symmetric matrices." No; the complex extension of real symmetric matrices would be symmetric complex matrices. Why conjugate everything across the diagonal? — Preceding unsigned comment added by Philgoetz (talk • contribs) 20:57, 12 November 2024 (UTC)

Personally, I often find it useful, whenever a product like ⁠${\bar {z}}w$⁠ (or ⁠$z{\bar {w}}$⁠) appears, to think of these numbers as representing two-dimensional Euclidean vectors, and this kind of product then represents the geometric product with real and imaginary parts representing the scalar-valued dot product and the bivector-valued wedge product of the two vectors, respectively. Symbolically, ⁠${\bar {z}}w=\operatorname {Re} ({\bar {z}}w)+i\operatorname {Im} ({\bar {z}}w)$⁠ stands in for ⁠$uv=u\cdot v+u\wedge v$⁠, or split into coordinates, ⁠$(a-bi)(c+di)=(ac+bd)+(ad-bc)i$⁠ stands in for ⁠$(a\mathbf {e} _{1}+b\mathbf {e} _{2})(c\mathbf {e} _{1}+d\mathbf {e} _{2})=(ac+bd)+(ad-bc)\mathbf {e} _{12}$⁠. Under this interpretation, complex numbers are being used because the geometric product and multivectors are an undertaught/underused mathematical formalism, despite being arguably more appropriate for the context, so complex numbers are a convenient stand-in with wider adoption that can sort of be made to work if you squint at them. –jacobolus (t) 04:49, 13 November 2024 (UTC)