We give a necessary and sufficient condition that a singular square matrix A A over an arbitrary field can be written as a product of two matrices with prescribed eigenvalues. Except when A A is a 2 × 2 2 \times 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 A . We use this result to prove results about products of hermitian and positive semidefinite matrices simplifying and strengthening some known results.