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I am used to thinking of matrices in terms of linear transformations, but it occurred to me that skew-symmetric matrices are a potential counterexample.

I can think of at least two examples in which skew symmetric matrices crop up:

  • As a representation space for $\wedge^2(V)$ for some vector space $V$. One of presumably very many examples of this occurs on pp. 226-227 of Bhargava's Higher Composition Laws I; one interesting feature of this construction is that the determinant of the matrix you get is interesting, being the square of a naturally occuring polynomial equation of the matrix entries (the Pfaffian).

  • As the Lie algebra of the orthogonal group.

In neither case does one naturally regard $n \times n$ skew-symmetric matrices as acting by linear transformations on an $n$-dimensional vector space. Is there a case where one does?

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  • $\begingroup$ In fluid mechanics, the anti-symmetric part of the strain tensor does play a role. And it arises as a transformation of the form $$\mathbf{x}\mapsto \omega \times \mathbf{x},$$ where $\times$ is the vector product of $3$-dimensional space. Does it count as an example? $\endgroup$
    – Giuseppe Negro
    Commented Jul 22, 2013 at 16:26
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    $\begingroup$ The matrices in the Lie algebra of $O(n)$ act naturally on $\mathbb{R}^n$. When $n=3$ one obtains the situation described by Giuseppe Negro. $\endgroup$
    – Liviu Nicolaescu
    Commented Jul 22, 2013 at 16:35
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    $\begingroup$ I think that the essential question is what you mean by 'naturally'. For a general vector space $V$ over a field, there is nothing natural about skew-symmetric matrices, as this condition is not preserved by an arbitrary choice of basis (which is how one gets a 'natural' correspondence between linear transformations and matrices in the first place). However, for a vector space $V$ endowed with a non-degenerate symmetric inner product that can be written as a sum of squares, the choice of an orthonormal basis for the inner product produces the natural correspondences you mention. $\endgroup$
    – Robert Bryant
    Commented Jul 22, 2013 at 17:09
  • $\begingroup$ Thanks to all: @Liviu: Guiseppe's example is very nice, but I understand that the cross product exists only in 1, 3, and 7 dimensions? Do you just mean in an infinitesimal sense, or would one ever "naturally" write down the action of a matrix in the Lie algebra of $O(n)$ on $\mathbb{R}^n$? If the latter, I would be very grateful if you would explain further. $\endgroup$
    – Frank Thorne
    Commented Jul 24, 2013 at 14:24
  • $\begingroup$ @Robert: Thank you very much. I guess I mean (roughly) an action that has some nice geometric interpretation, or which is studied in some other context. If I understood correctly, your last sentence is not answering my question in the affirmative? $\endgroup$
    – Frank Thorne
    Commented Jul 24, 2013 at 14:27

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