Geometric analysis of spectral stability of matrices and operators

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In this thesis, an attempt is made to undertake a systematic analysis of the sensitivity of eigen systems in the natural geometric framework of the spectral portraits of the matrices. The e-spectra and the spectral portraits are shown to be efficient graphical tools for sensitivity analysis of eigenvalues and spectral decomposition of matrices. The notion of e-spectra is also shown to be an appropriate logical setting for spectral analysis of matrices which are known only up to a given accuracy. The geometric separation of eigenvalues of a matrix A which can be read off from the e-spectra of A is shown to be an appropriate measure of sensitivity of eigenvalues and spectral decompositions. For the l 2–norm, a characterization of the sensitivity of spectral decompositions is provided and in the process a problem raised by Demmel[13] is solved. Sufficient conditions are obtained for the stability of spectral decompositions with respect to operator norms. Several bounds on the magnitude of the perturbations which ensure stability are also derived. Under appropriate assumptions, a conjecture of Demmel[13] on the separation of matrices is also settled.
Supervisor: Rafikul Alam