Refined Perturbation Bounds for Eigenvalues of Hermitian and Non-Hermitian Matrices
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SIAM Journal on Matrix Analysis and Applications 31 (1): 40-53 (January 2009)

We present eigenvalue bounds for perturbations of Hermitian matrices and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications include principal component analysis under a spiked covariance model, and pseudo-arclength continuation methods for the solution of nonlinear systems.
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