Thanks very much to all for their suggestions. I found the following very interesting paper, which shows how to go from the Weyl Inequalities to constructing a matrix having given eigenvalues & singular values which satisfy them. The paper does a great job of elucidating the elements of the Weyl proof. Chu, Moody T. "A Fast Recursive Algorithm for Constructing Matrices with Prescribed Eigenvalues and Singular Values" http://www4.ncsu.edu/~mtchu/Research/Papers/svdeig.pdf The paper also makes the statement: "It turns out, due to Horn[3], that the above conditions are also sufficient, i.e., (1.2) and (1.3) are the _only_ relations between the eigenvalues and the singular values of any general matrix." [Emphasis supplied]. My takeaway from this discussion is that singular values won't tell me very much about eigenvalues. At 09:18 AM 1/4/2010, Jean Gallier wrote:
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