Talk:Symmetric matrix

From Wikipedia, the free encyclopedia

	This article is within the scope of WikiProject Mathematics.
	Mathematics grading:	Start Class	High Importance	Field unassessed.

Shouldn't be better to create a distinct entry for 'skew-symmetric matrix' ?

[edit] Basis, Eigenvectors

It's easy to identify a symmetric matrix when it's written in terms of an orthogonal basis, but what about when it's not? Is a real-valued matrix symmetrix iff its eigenvectors are orthogonal? —Ben FrantzDale 00:31, 11 September 2006 (UTC)

Reading more carefully answers my question: "Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix." So apparently the answer is yes. —Ben FrantzDale 15:27, 11 September 2006 (UTC)

I believe you're confusing a couple of concepts here. A matrix is a rectangular array of numbers, and it's symmetric if it's, well, symmetric. Of course, a linear map can be represented as a matrix when a choice of basis has been fixed. On the other hand, the concept of symmetry for a linear operator is basis independent. Greg Woodhouse 01:34, 30 November 2006 (UTC)

being symmetric with real entries implies unitarily diagonalizable; the converse need not be true. anti-symmetric matrices with real entries are normal therefore unitarily diagonalizable. but the eigenvalues are no longer real, so one must speak of unitary matrices, rather than orthogonal. Mct mht 04:07, 12 September 2006 (UTC)

[edit] Symmetric matrices are usually considered to be real valued

I've made several changes to indicate that symmetric matrices are generally assumed to be real valued. With this, the real spectral theorem can be stated properly. VectorPosse 05:03, 12 September 2006 (UTC)