A spectral approach to polynomial matrices solvents

Using the concept of Jordan chain of L(λ) = Alλl + Al−1λl−1 + … + A0, where the Ai are n × n matrices with complex entries, we obtain a necessary and sufficient condition for the existence of S ϵ C n×n such that L(S) = 0. A way of finding those matrices, usually called solvents of L(λ), is given as well.