On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials

Every matrix polynomial ${\mathbf{f}}_n$ can be written in the form $${\mathbf{f}}_n(z)=\mathbf{h}(z^2)+z{\mathbf{g}}_n(z^2)\mathrm{.}$$ The matrix polynomial ${\mathbf{f}}_{2m}$ is said to be of Hurwitz type if the expression ${\mathbf{g}}_{2m}(z){\mathbf{h}}_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial ${\mathbf{f}}_{2m+1}$ is of Hurwitz type if $\frac{1}{z}{\mathbf{h}}_{2m+1}(z){\mathbf{g}}_{2m+1}^{-1}(z)$ has the same property. We derive an explicit form of the Bezoutian associated with Hurwitz-type matrix polynomials. Using this explicit form, we provide an explicit proof that Hurwitz-type matrix polynomials are Hurwitz matrix polynomials. In [52], the Hurwitzness of Hurwitz-type matrix polynomials was also studied. Finally, we propose an extension of the class of Hurwitz-type matrix polynomials by adding to a non–Hurwitz-type matrix polynomial another matrix polynomial so that the resulting matrix polynomial is of Hurwitz type.