Talk:Skew-symmetric matrix

Corrections

The skew-symmetric n×n matrices form a vector space of dimension (n2 − n)/2. This is the tangent space to the orthogonal group O(n).

Shouldn't that say something like "the tangent space to the orthogonal group O(n) at the identity matrix"? Surely it's not the tangent space at all points. Josh Cherry 02:28, 12 Apr 2004 (UTC)

Yes, the tangent space at I. Charles Matthews 05:39, 12 Apr 2004 (UTC)

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Skew-symmetric matrices fall into the category of normal matrices and are thus subject to the spectral theorem, which states that any real or complex skew-symmetric matrix can be diagonalized by a unitary matrix.

This statement appears incorrect; a complex skew-symmetric matrix is not necessarily normal, hence the spectral theorem may not apply. Is there something I am missing?

You're right. I added the requirement that the matrix be real. Thanks for mentioning it. -- Jitse Niesen (talk) 05:09, 12 June 2006 (UTC)

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Skew-symmetric matrices form the tangent space to the orthogonal group O(n) at the identity matrix. In a sense, then, skew-symmetric matrices can be thought of as infinitesimal rotations.

This needs clarification. Skew-symmetric matrices are not rotation matrices at all. A small multiple of a skew-symmetric matrix may be *added* to the identity matrix to produce an approximation to a small rotation, but there is nothing especially infinitesimal about this. This works because sine is approximately linear near 0 and cosine is approximately constant near 0. The skew-symmetric matrix then defines the direction of rotation (in R3, the axis of rotation). The orthogonal (rotation) matrix produced by exponentiating a skew-symmetric matrix is a rotation in this direction (in R3, about this axis) which is scaled by the magnitudes of the non-zero elements.

Well, "infinitesimal" is used in this more-or-less metaphorical sense in many areas of mathematics. Leibniz thought of the curvature of a curve as being entirely absent when one looks at an infinitely short arc, so that f(x + dx) = f(x) + f'(x) dx. Those usages are vestiges of Leibniz's idea, no longer (usually) taken literally. Michael Hardy 21:54, 16 December 2006 (UTC)

The point I was trying to make was about thinking of general (finite, non-infinitesimal) skew-symmetric matrices as representing or corresponding to infinitesimal rotations. An infinitesimal skew-symmetric matrix (one where all components are multiplied by dx, say) can be thought of as an infinitesimal rotation, since it can used as a rotation directly, when added to the identity, or can produce by exponentiation a rotation matrix for an infinitesimal angle. A finite (non-infinitesimal) skew-symmetric matrix can be "thought of" as a rotation (it corresponds to one, the one you get from exponentiating it), but there is something else that must be brought in to relate this to an infinitesimal rotation. It's the connection between not specifically infinitesimal skew-symmetric matrices and specifically infinitesimal rotations I am commenting on.

Note that infinitesimal in this context has a precise technical meaning as generator of a (semi)group. The infinitesimal object belongs to a Lie Algebra, and the exponential map takes it to the corresponding Lie Group. --pma (talk) 09:55, 25 November 2009 (UTC)

Alternating form

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)

Outline for proof

The article page states that a proof is needed for "A is skew-symmetric if and only if x^TAx = 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: x^TAx = x^T(x^TA^T)^T = -x^T(x^TA)^T = -((x^TA)x)^T = -(x^TAx)^T = -x^TAx . This can only be satisfied if x^TAx = -x^TAx = 0. Utilized properties: A^T = -A (A is skew-symmetric) and (x^Tx)^T = x^Tx (scalars are symmetric). --Catskineater (talk) 16:58, 23 September 2008 (UTC)

Actually your proof also has an error I think. -(x^TAx)^T = -x^TA^Tx = x^TAx. Unless I'm missing something.Lizard86 (talk) 19:48, 15 October 2008 (UTC)

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)

Where you say -(x^TAx)^T = -x^TAx, 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)

ah, i see. x^TAx is a scalar quantity and like i said, scalars are symmetric, so (x^TAx)^T = x^TAx. But you're right in that this proves only one direction (namely "A skew-symmetric -> x^TAx=0").--Catskineater (talk) 23:02, 16 October 2008 (UTC)

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)

Actually it's a bit simpler than that... If x^TAx = 0 for all x then also (x+y)^TA(x+y) = 0 for all x and y; just expand it and you'll get x^TAy = -y^TAx = -x^TA^Ty for all x and y, so A=-A^T. Check if as I wrote it is clear enough. --pma (talk) 08:11, 10 November 2009 (UTC)

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, x^T 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)

Extra minus sign?

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)

Yes, incorrect. One finds, for any A and B: [A,B]^T = [B^T,A^T], and if both are skew-symmetric this is also = [B,A]=-[A,B], meaning that [A,B] is skew-symmetric.--pma (talk) 08:41, 23 November 2009 (UTC)

Yeah, it looks like a typo. If the commutator of two skew-symmetric matrices were again skew symmetric then we would have [A,B]^T = −[A,B], and indeed: [A,B]^T = (AB − BA)^T = B^TA^T − A^TB^T = BA − AB = −[A,B]. I've just rewritten the proof to make it more direct. ~~ Dr Dec (Talk) ~~ 20:21, 24 November 2009 (UTC)

Simple proof of equation $A=O^{T}\Sigma O$

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||A^T - λE|| = det||- A + λE||). Consider A². This matrix is symmetric. It can be diagonalized by rotation. Thus O^TA²O = C. The diagonal elements of C are real, negative, and double-degenerate. O^TAO = 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)

Clarification for spectral theory section

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)

Octonions

Is it worth mentioning that the matrix formed by multiplication of two octonions $e_{i}e_{j}$ 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)

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)

Drop-down list when searching

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)

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)

Block diagonal form via Real Schur decomposition

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)