The Hadamard Factorization of Hurwitz and Schur stable Polynomials

Abstract

We consider the Hadamard (i.e. the coefficient-wise) product of two poly nomials. The set of the Hurwitz stable polynomials is closed under the Hadamard product, whereas the set of the Schur stable polynomials is not. In this note we show that each Schur stable polynomial allows a Hadamard factorization into two Schur stable polynomials, whereas there are Hurwitz stable polynomials of degree 4 which do not have a Hadamard factorization into two Hurwitz stable polynomials of degree 4.