Upper triangularization of matrices by permutations and lower triangular similarity transformations (Q1109521)

Let A be a complex \(m\times m\) matrix and let \(\alpha_ 1,...,\alpha_ m\) be an ordering of the eigenvalues of A. Then there exist a permutation matrix P and a lower triangular matrix L such that \((PL)^{-1}APL\) is upper triangular with diagonal \((\alpha_ 1,...,\alpha_ m)\). The proof is constructive, two examples are given. The result can be applied in systems theory.