Talk:Conjugate transpose

One more way of denoting the conjugate transpose?
Why aren't "*T" listed as a possible notation for the conjugate transpose? This notation, which I've previously encountered in statistical signal processing, is presumably more intuitive, since it really is just a combination of conjugate and transpose. --Fredrik Orderud 00:57, 21 Apr 2005 (UTC)


 * I never encountered the notation as you say, $$A^{*T}\,$$. I'd say it looks a bit clumsy. Besides, note that in this article (and in many other places) they use the bar for the conjugate, so then your notation might need to be changed to $${\overline A}^{T}$$.


 * If you think the notation you mention is in widespread use, and its addition would be helpful, you could add it to the list of alternative notations, next to     $$A^*, A^H, \mbox{ or } A^\dagger\,$$.  But again, the * thing might mean different things to different people. Oleg Alexandrov 01:47, 21 Apr 2005 (UTC)


 * Ok, the notation may very vell not be in widespread use. I've previously also used the "H" notation (in medical ultrasound). What about including the $$ A^H \equiv {\overline A}^{T} $$ (or simmilar using the "*" notation) equation, which I find easier ut understand than the current definition? --Fredrik Orderud 09:07, 21 Apr 2005 (UTC)


 * Sure. But again, putting the notation with the star where star means conjugate, will conflict with the star where star means transpose. The notation with $$ {\overline A}^{T} $$ looks fine. Probably it is more appropriate to include it in the list of alternative notations, than to replace everywhere the current notation with this one. What do you think? Oleg Alexandrov 15:27, 21 Apr 2005 (UTC)


 * I added the $$ A^H \equiv {\overline A}^{T} $$ definition since it seemed like that was ok.


 * I added the A' notation because it is used in a lot of books and papers. For some history see:

https://math.stackexchange.com/questions/2582286/who-established-the-tradition-of-using-x-prime-instead-of-xt-to-denote Jeff Fessler — Preceding undated comment added 13:33, 30 October 2022 (UTC)

Rant
Start Rant. i just wanted to say that this maths article is yet another overly jargon filled maths wikipedia page. I know this is a general gripe but I find these maths pages seem to be written for the use of some one who already understands the ideas, which is completely the opposite idea behind the whole reason for having a wikipedia. that said, i still don't know how to fix the problem. i spose the problem is that the people qualified to write about these sorts of things are mathematicians who have mentally digested the jargon a long time ago and dont have to think about them. also there is this whole 'minimalism' thing in maths where if you show a proof or something you only show the bare minimum number of steps such that a laymen often can't follow the proof all the way through because an 'assumed' step has been ommitted.

but this isn't a proof, this is wikipedia, a platform on which people can share ideas and so omitting 'assumed' steps is probably a bad thing. It limits the audience, makes the person who can't understand it feel swamped and dumb (yes im speaking from personal experience ;-P ) and from my perspective makes the writer of the 'proof' or explanation look elitist. oh well. End Rant. -anon


 * One thing which would go towards a fix of this issue is if people like you were more specific. What parts do you have problems with? As you say, many people writing maths articles already know the subject, which makes it hard to see what parts are problematic. -- Jitse Niesen (talk) 13:49, 17 February 2006 (UTC)
 * Exactly. Which "proof" do you have a problem with? There aren't even any proofs on the page as far as I can tell. --King Bee 14:41, 15 November 2006 (UTC)

No joke. I have been searching forever to just figure out what U*U=UU*=I was referring to mentioned everywhere else. A quick easy 2x2 matrix multiplied by another 2x2 matrix with a link to examples of how a matrix is multiplied. We need links to complex symbols explained down to the most very basic terms. Wikipedia is extremely bad at drilling down basic easy concepts like this. Perhaps a link to U* to explain exactly how this property works. The faster more people understand things, the faster society will progress. examples like this: http://tutorial.math.lamar.edu/Classes/LinAlg/PropsOfMatrixArith.aspx

Other meaning of Adjoint
Hiya, there should be a link here to Adjugate Matrix. 'll do it if no-one else does, just don't know how yet. Herman Stel 23:10, 14 November 2006 (UTC)


 * Did you see that there already is a link to adjugate matrix, at the end of the section "Basic remarks"? -- Jitse Niesen (talk) 00:25, 15 November 2006 (UTC)

Conventional depiction of matrices
In the literature, matrix variables are usually upright Roman boldface. This page needs editing to fit the standard. The overbar for conjugation is a new one on me. —The preceding unsigned comment was added by 200.122.159.125 (talk) 01:30, 28 December 2006 (UTC).


 * That depends very much on which literature you are familiar with. Both italics for matrices and overbar for conjugation are common in the mathematical literature. So I changed it back. -- Jitse Niesen (talk) 13:30, 28 December 2006 (UTC)

Shouldn't the comma between the subscripts in the first equation be ommitted? Scot.parker 11:32, 23 July 2007 (UTC)

Another way to find the adjoint of a transformation?
An adjoint of a transformation is just one for which the inner product $$\langle Av,w\rangle = \langle v, A^* w \rangle$$. But for a transformation in terms of a basis which isn't orthonormal, it's not so simple as a conjugate transpose. Check for yourself. --68.161.190.195 (talk) 07:09, 18 December 2007 (UTC)

Incorrect Definition
The page's definition ($$A^* = (\overline{A})^\mathrm{T} = \overline{A^\mathrm{T}}$$) is lacking: is it only true when we are working with orthonormal bases. The following is an excerpt from Chapter 6, page 120-1 of "Linear Algebra Done Right" (by Sheldon Axler):

"The proposition shows how to compute the matrix of T* from the matrix of T. Caution: the proposition below applies only when we are dealing with orthonormal bases -- with respect to nonorthonormal bases, the matrix of T* does not necessarily equal the conjugate transpose of the matrix of T." —Preceding unsigned comment added by Smithg86 (talk • contribs) 22:31, 18 May 2008 (UTC)


 * In that quote, T is a linear operator and T* denotes the adjoint of the operator. -- Jitse Niesen (talk) 16:33, 26 November 2008 (UTC)

Eigenvectors & conjugate transposition
Is there anything known about the relation between eigenvectors of A and eigenvectors of its conjugate transpose? I know they are equal if A is diagonalisable, but is there something known about the non-diagonalisable case (other than: left & right singular vectors are exchanged)? Catskineater (talk) 19:31, 30 April 2010 (UTC)

Motivation Section
In the Motivation section, the matrix shown is not skew symmetric. I'm not sure what it should be called. John Yoepp —Preceding unsigned comment added by 134.223.116.200 (talk) 22:28, 7 December 2010 (UTC)

I removed the claim about a complex number being representable as a skew-symmetric matrix. However, the paragraph might be rewritten. You can define the complex number as a sum of a diagonal matrix (the real part) and a skew-symmetric matrix (the imaginary part). And it would of course be true for a pure imaginary number. Andy (talk) 16:49, 9 July 2011 (UTC)

Better citation for notation
Wolfram MathWorld is not a reliable source. I have often found unsupported opinions and outright mistakes in it. As a reference for notations for the complex conjugate, a real book would be highly desirable. Zaslav (talk) 11:27, 1 December 2023 (UTC)