FACTORIZATION OF SINGULAR MATRICES

We give a necessary and sufficient condition that a singular square matrix A over an arbitrary field can be written as a product of two matrices with prescribed eigenvalues. Except when A is a 2 x 2 nonzero nilpotent, the condition is that the number of zeros among the eigenvalues of the factors is not less than the nullity of A. We use this result to prove results about products of hermitian and positive semidefinite matrices simplifying and strengthening some known results.