On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials

Abstract: Every matrix polynomial $\f_n$ can be written in the form $\f_n(z)=\h_n(z^2)+z\,\g_n(z^2)$. The matrix polynomial $\f_{2m}$ is said to be of Hurwitz-type if the expression $\g_{2m}(z)\h_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial $\f_{2m+1}$ is of Hurwitz type if $\frac{1}{z}\h_{2m+1}(z)\g_{2m+1}^{-1}(z)$ has the same property. Using orthogonal matrix polynomials on the interval $[0,+\infty)$ and their associated second-kind polynomials, we establish the coprimeness of the matrix polynomial pairs $(\h_{2m},\g_{2m})$ and $(\h_{2m+1},z\g_{2m+1})$. Assuming a commutativity-type condition on the coefficients of $f_n$, we derive an explicit form of the Bezoutian associated with Hurwitz-type matrix polynomials. Under the same assumption, we also show that Hurwitz-type matrix polynomials are indeed Hurwitz matrix polynomials. Finally, we propose an extension of the class of Hurwitz-type matrix polynomials.