Eigenvalues of products of matrices and submatrices in certain positivity classes

If A and B are n-by-n, positive semidefinite Herimitian matrices, then lambda (min)(AB) less than or equal to lambda (min)(A[alpha ]B[alpha]), for any phi not equal alpha subset of or equal to {1, 2, ..., n}. A certain converse is given, as well as analogs for the product of several M-matrice s and totally nonnegative matrices. However, analogs for lambda (max) (othe r than the case of totally nonnegative matrices) and for singular values of general matrices, etc., fail.