Jean writes: << The non-negative square roots, sigma_1, ..., sigma_n, of the eigenvalues of M.M' are indeed known as the singular values of M. The Singular Value DecompositionTheorem (SVD) says that M = U Sigma V', for some unitary matrices, U , V , and where Sigma = diag(sigma_1, ..., sigma_n) (I assume M is an n X n matrix).
The Singular Value Decomposition has an interesting interpretation over the reals. Suppose L_1 and L_2 are linear n-dimensional subspaces of R^k. If n = 1 their relative position in R^k is described by one number, the angle between these two lines (or its cosine). Let {u_1,...u_n} and {v_1,...v_n} be orthonormal bases for L_1 respectively L_2 in R^k. Let U and V be the k x n matrices whose columns are the u_j's respectively the v_j's. Then the n x n matrix A := U^t V has its ij^th entry just <u_i, v_j> (dot product). The singular value decomposition of A is SVD = P^t A Q, where SVD is a diagonal matrix of nonnegative entries in descending order, and P and Q are orthogonal. This defines SVD uniquely. Back to the geometry of L_1 and L_2: Among all the unit vectors x in L_1 and y in L_2, there is an x and y that maximize the dot product <x,y>, i.e., minimize their angle. (It may happen that the angle is 0.) This dot product will be the first entry of the SVD matrix S. Now let L_1' and L_2' be the orthogonal complements of x respectively y in L_1 and L_2. Again let x' and y' be the vectors of L_1' and L_2' respectively having the least angle. Then <x',y'> is the second entry of SVD. Etc. The sequence of vectors x, x', x'',... is orthonormal, as is y, y', y'',.... The x's are given by the columns of the n x k matrix VQ, and likewise for the y's and UP. The sequence of (nonnegative) cosines, i.e., the singular values, on the diagonal of SVD completely characterize the relative positions of L_1 and L_2 up to an isometry of R^k. The angles (inverse cosines of the singular values) are called the principal angles of L_1 and L_2. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele