Talk:Skew-symmetric matrix

message: The definition of alternating form here doesn't agree with those at alternating form and bilinear form, which I believe to be the usual one.  I appreciate that the caveat characteristic not two is in place.  Richard Pinch (talk) 06:49, 30 June 2008 (UTC)
thread_title:  Alternating form 

message: The article page states that a proof is needed for "A is skew-symmetric if and only if xTAx = 0 for all real vectors x". I don't have the time or editing experience to write a sound proof, but the basis idea goes like this: xTAx = xT(xTAT)T = -xT(xTA)T = -((xTA)x)T = -(xTAx)T = -xTAx . This can only be satisfied if xTAx = -xTAx = 0. Utilized properties: AT = -A (A is skew-symmetric) and (xTx)T = xTx (scalars are symmetric). --Catskineater (talk) 16:58, 23 September 2008 (UTC)
replies: text: As far as i can see, you basically wrote the same as me. What part do you think is erroneous?--Catskineater (talk) 01:54, 16 October 2008 (UTC)
replies: text: Where you say -(xTAx)T = -xTAx, at the end of the long chain of equalities. Also, your proof only does one direction of the if and only if. -- Jitse Niesen (talk) 10:58, 16 October 2008 (UTC)
replies: text: ah, i see. xTAx is a scalar quantity and like i said, scalars are symmetric, so (xTAx)T = xTAx. But you're right in that this proves only one direction (namely "A skew-symmetric -> xTAx=0").--Catskineater (talk) 23:02, 16 October 2008 (UTC)
replies: text: Okay, I put the proof back in, with some explanation. The other direction seems more difficult (I can think of a proof using the real Jordan form but it's a bit too long to put in here and also probably Original Research), so it would be good to have a reference for it. -- Jitse Niesen (talk) 11:34, 17 October 2008 (UTC)
replies: text: Actually it's a bit simpler than that... If xTAx = 0 for all x then also (x+y)TA(x+y) = 0 for all x and y; just expand it and you'll get xTAy = -yTAx = -xTATy for all x and y, so A=-AT. Check if as I wrote it is clear enough. --pma (talk) 08:11, 10 November 2009 (UTC)

text: The proof is only one way, since the converse is not true. As an example, take x = (1,1) and A = diag(-1,1). Clearly, xT A x = 0, but A is not skew-symmetric.  —Preceding unsigned comment added by 131.114.31.18 (talk) 08:46, 29 July 2010 (UTC)
thread_title:  Outline for proof 

message: It says here that:
    It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric, using the fact that
    [A,B]T = - [BT,AT]
Isn't that minus sign incorrect?  —Preceding unsigned comment added by Skaphan (talk • contribs) 00:20, 23 November 2009 (UTC)
replies: text: Yeah, it looks like a typo. If the commutator of two skew-symmetric matrices were again skew symmetric then we would have  and indeed:  I've just rewritten the proof to make it more direct. ~~ Dr Dec (Talk) ~~  20:21, 24 November 2009 (UTC)
thread_title:  Extra minus sign? 

message: Eigenvalues of A are imaginary numbers, because iA is Hermitian matrix, and the eigenvalues of Hermitian matrices are real, as is known e.g. from the course of quantum mechanics. Eigenvalues appear in pairs ±λ due to the antisymmetry of A 
(det||A - λE|| = det||AT - λE|| = det||- A + λE||). Consider A2. This matrix is symmetric. It can be diagonalized by rotation. Thus OTA2O = C. The diagonal elements of C are real, negative, and double-degenerate. OTAO = Σ is square root of C. In the field of real numbers the square root can be extracted uniquely up to a sign. So, we get a canonical representation of the text. Trompedo (talk) 06:49, 16 January 2013 (UTC)
thread_title:  Simple proof of equation <math>A = O^T \Sigma O </math> 

message: In the decomposition  A = Q \Sigma Q^T  it should be clarified that the eigenvalues of  \Sigma  are not the same as the eigenvalues of  A . The use of the same  \lambda  in the first paragraph and the one after can cause confusion.  — Preceding unsigned comment added by 192.17.100.34 (talk) 12:35, 23 August 2017 (UTC)
thread_title:  Clarification for spectral theory section  

message: Is it worth mentioning that the matrix formed by multiplication of two octonions e_ie_j is skew-symmetric (and that multiplication across the octonions is alternative, in line with the section on §Skew-symmetric and alternating forms, as long as I'm not getting alternative algebras and alternating forms muddled up), or is this a trivial property? — Sasuke Sarutobi (talk) 14:02, 29 September 2017 (UTC)
replies: text: Ignore me. I've just realised that it's not skew-symmetric where i=0 or j=0. Apologies. — Sasuke Sarutobi (talk) 14:16, 29 September 2017 (UTC)
thread_title:  Octonions 

message: On the Wikipedia home page, when you search for "skew-symmetric", the summary that appears in the drop-down list for 'skew-symmetric matrix' summarises it as, 'square matrix whose inverse equals it's negative'. This should say '...transpose equals...'. I can't find where this text is taken from. Is there somewhere it can be edited, or is it some bot generated text that's being cached somewhere? -- PSmythirl (talk) 09:49, 13 April 2020 (UTC)
replies: text: See Wikipedia:Short description. That entry was taken from the "Description" field in the Wikidata entry for the article Skew-symmetric matrix, see Wikipedia:Wikidata. I think I have fixed the short description now. Thank you for noticing that mistake. – Tea2min (talk) 17:36, 13 April 2020 (UTC)
thread_title:  Drop-down list when searching 

message: The Real Schur decomposition yields the stated form with 2x2 block diagonals and orthogonal transformation matrix, though practically (numerical implementations) it may not have them sorted in the stated order. I think this ought to be mentioned though I can't find a good source for it (the Schur decomposition article doesn't even mention the real form). --Nanite (talk) 06:33, 15 March 2024 (UTC)
thread_title:  Block diagonal form via Real Schur decomposition