Ion Zaballa "On the sign characteristic of selfadjoint quadratic matrix polynomials"

A variety of applications lead to the spectral analysis of quadratic matrix polynomials with Hermitian matrix coefficients. In this talk we present and discuss canonical forms for these polynomials under the action of - both - congruence and similarity transformations of the poly- nomials and their linearizations. Canonical Jordan structures will be clarified.

We bring together two lines of attack: (a) analytic via direct reduction of the system itself by parameter dependent unitary similarity, and (b) algebraic via reduction of symmetric linearizations of the system by congruence transformations.

Complete descriptions of the associated canonical structures are provided These canonical structures include the so-called sign characteristic. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics will be presented and connections among them will be clarified. Most of them are base-dependent. We will show which linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory and how to obtain the corresponding canonical form. In particular, the case of selfadjoint quadratic matrix polynomials with singular leading coefficient will be considered. The general theory will be illustrated through examples and graphics.

This is joint work with Peter Lancaster.

Sponsors