Strong linearizations of polynomial and rational matrices and recovery of spectral data

Abstract

Linearization is a classical technique widely used to deal with matrix polynomial. The main purpose of the thesis is to construct and analyze strong linearizations of polynomial and rational matrices. The first part of the thesis is devoted to construction of strong linearizations of matrix polynomials including structure-preserving strong linearizations and the recovery of eigenvectors, minimal bases and minimal indices of matrix polynomials from those of the linearizations. The second part of the thesis is devoted to construction of strong linearizations of rational matrices including structure-preserving strong linearizations and the recovery of eigenvectors, minimal bases and minimal indices of rational matrices from those of the linearizations.Fiedler pencils (FPs), generalized Fiedler pencils (GFPs), Fiedler pencils with repetition (FPRs) and generalized Fiedler pencils with repetition (GFPRs) are important family of strong linearizations of matrix polynomials which have been studied extensively over the years. It is well known that the family of GFPRs of matrix polynomials subsumes the family of FPRs and is an important source of strong linearizations, especially structure-preserving strong linearizations of structured matrix polynomials. We propose a unified framework for analysis and construction of a family of Fiedler-like pencils, which we refer to as extended GFPRs (EGFPRs), that subsumes all the known classes of Fiedler-like pencils such as FPs, GFPs, FPRs and GFPRs of matrix polynomials.