On the relation between Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions

Abstract

In this paper, we elaborate on the relationship between the Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions. Our strategy is that, for a monic matrix polynomial, we associate a rational matrix-valued function with its even–odd split and then check the Hurwitz stability of the matrix polynomial by testing the Stieltjes property of the related rational matrix-valued function. On the basis of this relationship, we present matrix generalizations of a classical stability criterion by Gantmacher, Chebotarev theorem, Grommer theorem and some aspects of the modified Hermite–Biehler theorem. Our work is motivated by one of the authors’ recent stability studies linked with matricial Markov parameters.