Algebraic Linearizations of Matrix Polynomials

Abstract

We show how to construct linearizations of matrix polynomials $z\mathbf{a}(z){\mathbf{d}}_0+{\mathbf{c}}_0$, $\mathbf{a}(z)\mathbf{b}(z)$, $\mathbf{a}(z)+\mathbf{b}(z)$ (when $\mathrm{deg}\left(\mathbf{b}(z)\right)<\mathrm{deg}\left(\mathbf{a}(z)\right)$), and $z\mathbf{a}(z){\mathbf{d}}_0\mathbf{b}(z)+{\mathbf{c}}_0$ from linearizations of the component parts, $\mathbf{a}(z)$ and $\mathbf{b}(z)$. This allows the extension to matrix polynomials of a new companion matrix construction.

35 pages, 3 figures