Stability Theory for Matrix Polynomials in One and Several Variables with Extensions of Classical Theorems

TL;DR

This dissertation develops a stability theory for matrix polynomials, introducing hyperstability concepts and extending classical theorems like Gauss-Lucas and Szász inequalities to multivariable matrix contexts.

Contribution

The work introduces the concept of hyperstability for matrix polynomials and extends classical complex analysis theorems to multivariable matrix polynomial settings.

Findings

Extended Gauss-Lucas theorem to multivariable matrix polynomials

Derived Szász-type inequalities for matrix norms

Developed a new stability framework for matrix polynomials

Abstract

The file contains PhD Dissertation by Oskar Jakub Szyma\'nski. This work ends his study at Doctoral School of Exact and Natural Sciences at Jagiellonian University where the Author has attended in years 2019-2024. The subject of the Thesis is the stability theory developed for matrix polynomials. The main concept described in the Dissertation is hyperstability with respect to an open or closed subset of the complex plane. The notion of hyperstability is a contribution of the Author of the Thesis. Disseration is devoted mostly to regular polynomials, singular polynomials play marginal role in the Thesis. The Author of the Thesis has discovered several interesting results extending classical theorems from complex analysis. These are for example: multi-variable Gauss-Lucas theorem for matrix polynomials and Sz\'asz-type inequalities for matrix norms such as the matrix two-norm and the Frobenius norm.