This paper is concerned with the following questions. Given a square matrix A, when does there exist an invertible lower triangular matrix L such that L-1AL is upper triangular ? And if so, what can be said about the order in which the eigenvalues of A may appear on the diagonal of t-1AL ? The motivation for considering these questions comes from systems theory. In fact they arise in the study of complete factorizations of rational matrix functions. There is also an intimate connection with the problem of complementary triangularization of pairs of matrices discussed in [4].