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Relative Perturbation Theory: (I) Eigenvalue Variations

Authors:
Li, Ren-Cang
Technical Report Identifier: CSD-94-855
December 1994
CSD-94-855.pdf
CSD-94-855.ps

Abstract: In this paper, we consider how eigenvalues of a matrix A change when it is perturbed to ~A = D1AD2 and how singular values of a (nonsquare) matrix B change when it is perturbed to ~B = D1BD2, where D1 and D2 are assumed to be close to unitary matrices of suitable dimensions. We have been able to generalize many well-known perturbation theorems, including Hoffman-Wielandt theorem and Weyl-Lidskii theorem. As applications, we obtained bounds for perturbations of graded matrices in both singular value problems and nonnegative definite Hermitian eigenvalue problems.