Talk:Skew-symmetric matrix

From Wikipedia, the free encyclopedia

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)

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)

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.