I think Jean has accurately nailed the source of my confusion ---- failing to take into account the constraint that S must be positive (semi-)definite. Note that it is quite tricky to determine from some (rather casual) statements of this result exactly which are constraints and which are properties! On 9/9/10, Jean Gallier <jean@cis.upenn.edu> wrote:
Dear Fred,
The confusion has to do with the fact that there are two polar decompositions:
M = QS_1 M = S_2Q
where Q is orthogonal and S_1, S_2 are symmetric, POSITIVE semi-definite. In general, S_1 \not= S_2.
The matrix M = [[0, 1], [1, 0]] has eigenvalues +1 and -1, so it is not positive and only the decomposition M I is acceptable.
If M is invertible, then these decompositions are unique. If M is normal, then S_1 = S_2. Proofs can be found as early as in Chevalley, Theory of Lie Groups , 1946. I have a proof in my book (Chapter 12, Thm 12.1.3): http://www.cis.upenn.edu/~cis610/geombchap12.pdf
I hate to say this but Ken Shoemake (who was once a student at UPenn is not a very reliable source.
Best,
-- Jean Gallier _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun