On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

This paper is a direct continuation of the author's recent investigations [4] on the non-degenerate truncated matricial Stieltjes problem. Inspired by a characterization of scalar Hurwitz polynomials via continued fractions (see e.g.  Gantmacher [15]) a class of matrix Hurwitz type polynomials is introduced. It is shown that this class is intimately related with Stieltjes positive definite sequences of matrices and can be expressed in terms of the associated Stieltjes quadruple of sequences of left orthogonal matrix polynomials. It is shown that a monic matrix polynomial is a matrix Hurwitz type polynomial if and only if the sequence of its Markov parameters is a Stieltjes positive definite sequence.