[math-fun] polar decomposition

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