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). A non-trivial relationship between the eigenvalues, lambda_1, ..., lambda_n, of M, and its singular values, sigma_1, .., sigma_n, assuming that | lambda_1 | \geq | lambda_2 | \geq ... \geq | lambda_n | and sigma_1 \geq sigma_2 \geg ... \geq sigma_n, are the Weyl Inequalities (1949): | lambda_1 | ... | lambda_k | \leq sigma_1 ... sigma_k, for k = 1, .., n - 1 and | lambda_1 | ... | lambda_n | = sigma_1 ... sigma_n. See Horn and Johnson, Topics in Matrix Analysis, page 171. Best, -- Jean Gallier