Localizing Spectra and Pseudospectra of Matrices and Matrix Polynomials
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2022
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Abstract
This thesis considers different aspects of the study of sets that localize spectra and
pseudospectra of matrices and matrix polynomials. Given a block upper triangular
matrix, the literature has several results for outer approximations of the pseudospec-
tra of the matrix by the pseudospectra of the diagonal blocks. The thesis provides
inner approximations of the pseudopectra of a block upper triangular matrix in terms
of pseudospectra of the diagonal blocks. Next the definitions of block Geršgorin sets,
block Brualdi sets, block minimal Geršgorin sets and permuted pointwise minimal
Geršgorin sets are extended to matrix polynomials in homogeneous form. The use
of the homogeneous form is justified by its unified treatment of both the finite and
infinite eigenvalues. Many properties of these sets are derived and efficient numerical
methods for plotting the sets that can compare favourably with some existing meth-
ods under certain conditions, are proposed. In particular, the proposed methods may
be used to plot all eigenvalue localization sets for matrix polynomials without laying
a grid on any part of the complex plane.
Eigenvalue problems associated with the quadratic matrix polynomials have a
wide range of applications. Localizations of eigenvalues of such polynomials are
proposed via block Geršgorin sets that arise from some special linearizations. Several
properties of these sets are derived and used to obtain some easily computable bounds
on the eigenvalues of the matrix polynomial. They are also used to obtain sufficient
conditions on the coefficient matrices of the polynomial for its eigenvalues to lie in
particular regions of the complex plane that are important from the point of view
of applications. The results also include various structured matrix polynomials. As
an outcome of the analysis, several upper bounds on solutions of important distance
problems associated with structured and unstructured quadratic matrix polynomials
are derived for various choices of norms. In all cases, numerical experiments are
performed to illustrate the results.
Description
Supervisor: Bora, Shreemayee
Keywords
Pseudospectra of matrices, Block Geršgorin Loacalizations, Structured Matrix Polynomials