Definite Matrix Polynomials and their Linearization by Definite Pencils

Hyperbolic matrix polynomials are an important class of Hermitian matrix polynomials that contain overdamped quadratics as a special case. They share with definite pencils the spectral property that their eigenvalues are real and semisimple. We extend the definition of hyperbolic matrix polynomial in a way that relaxes the requirement of definiteness of the leading coefficient matrix, yielding what we call definite polynomials. We show that this class of polynomials has an elegant characterization in terms of definiteness intervals on the extended real line and that it includes definite pencils as a special case. A fundamental question is whether a definite matrix polynomial $P$ can be linearized in a structure-preserving way. We show that the answer to this question is affirmative: $P$ is definite if and only if it has a definite linearization in $\mathbb{H}(P)$, a certain vector space of Hermitian pencils; and for definite $P$ we give a complete characterization of all the linearizations in $\mathbb{H}(P)$ that are definite. For the important special case of quadratics, we show how a definite quadratic polynomial can be transformed into a definite linearization with a positive definite leading coefficient matrix—a form that is particularly attractive numerically.