Linealització de matrius polinomials regulars quadrades

Abstract

[en] The usual way to numerically find the spectral structure of a regular matrix polynomial is to linearize it and then calculate eigenvalues and eigenvectors using numerical techniques for pencils of matrices. The aim of this project is to study linearization of square matrix polynomials that are regular. Given a matrix polynomial of degree $l$, linearization involves finding an equivalent one of degree 1. In order to find an equivalent matrix pencil, the case in which matrix polynomials are monic with the First Companion Matrix as a linearization is first discussed. Moreover, in the nonmonic case, a generalised First Companion Matrix is found to be a linearization. Taking these into account, numerous linearizations are determined. To sum up applications such as calculating the resolvent form, the inverse problem and obtaining solutions of differential and difference equations are seen.