Localizing Spectra and Pseudospectra of Matrices and Matrix Polynomials

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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.
Supervisor: Bora, Shreemayee
Pseudospectra of matrices, Block Geršgorin Loacalizations, Structured Matrix Polynomials