Pseudospectra of Matrix Pencils and their applications in perturbation analysis of Eigenvalues and Eigendecompositions

The main theme of the thesis revolves around pseudospectra of matrix pencils and matrix polynomials and their applications in perturbation theory. More specifically, first, we develop a general framework for defining and analyzing pseudospectra of matrix pencils and matrix polynomials. The framework so developed unifies various definitions of pseudospectra of matrix pencils proposed in the literature. We dispel the perception that there are many nonequivalent ways of defining pseudospectra of matrix pencils/polynomials and put the analysis of pseudospectra of matrix pencils/polynomials on the same footing as that of matrices. Second, we analyze various properties of backward error functions associated with matrix pencils/polynomials. We introduce a notion of critical points of backward error functions and show that a critical point is a multiple eigenvalue of an appropriately perturbed pencil/polynomial. We now show that common boundary points of the components of pseudospectra of a matrix pencil/polynomial are critical points. We show that a minimal critical point can be read off from the pseudospectra of matrix pencils/polynomials. Hence a solution of Wilkinson’s problem for matrix pencils/polynomials can be read off from the pseudospectra of matrix pencils/polynomials. Given a diagonal pencil with distinct eigenvalues, we provide a simple procedure for the construction of nearest defective pencils. Third, we provide various pseudospectra inclusions for matrix pencils and show that pseudospectra inclusions can be gainfully used for analyzing stability of eigendecompositions. We introduce analogues of various notions of separation of matrices to the case of matrix pencils and show their utility in analyzing stability of eigendecompositions. We show that the separations such as sep, sepλ and gsep can be defined and analyzed on the same lines as that of matrices. Fourth, we present a general framework for the sensitivity analysis of eigenvalues of matrix pencils and matrix polynomials. We lay bare the big picture that lies behind the notion of sensitivity of eigenvalues of matrix pencils/polynomials. We show that

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