Bounds for the Eigenvalues of Matrix Polynomials

Abstract

Developing bounds for the eigenvalues of matrix polynomials is an interesting problem which has a lot of applications. In this thesis, several known bounds for the eigenvalues of matrix polynomials are presented. In addition, we derive new bounds for the eigenvalues of matrix polynomials with commuting coefficients. These bounds are based on norms, numerical radius, and spectral radius of the coefficient matrices. Various tools are used in the derivations, such as Frobenius companion matrix, the numerical radius inequalities, and matrix norms. In general, it is not possible to compare the sharpness of these bounds analytically. Therefore, we compare our new bounds with each other and with other known bounds numerically through a set of examples.