engleski

Perturbation of Partitioned Hermitian Generalized Eigenvalue Problem

This paper is concerned with Hermitian positive definite generalized eigenvalue problem $A-\lambda B$ for partitioned $$ A=[A_{; ; ; ; 11}; ; ; ; 0 \\ 0 & A_{; ; ; ; 22}; ; ; ; ] , B=[B_{; ; ; ; 11}; ; ; ; & 0 \\ 0 & B_{; ; ; ; 22}; ; ; ; ], $$ where both $A$ and $B$ are Hermitian and $B$ is positive definite. Bounds on how its eigenvalues varies when $A$ and $B$ are perturbed by Hermitian matrices. These bounds are generally of linear order with respect to the perturbations in the diagonal blocks and of quadratic order with respect to the perturbations in the off-diagonal blocks. The results for the special case of no perturbations in the diagonal blocks can also be used to bound the changes of eigenvalues of a Hermitian positive definite generalized eigenvalue problem after its off-diagonal blocks are dropped, a situation occurs frequently in eigenvalue computations. Presented results extend those of Li and Li ({; ; ; ; \em Linear Algebra Appl.}; ; ; ; , 395 (2005), pp.183--190). Another possible extension here is to derive quadratic eigenvalue perturbation bounds for diagonalizable matrix pencils with real spectra. Also established are perturbation bounds for a multiple eigenvalue to reflect the distinguished feature that its different copies may exhibit quite different sensitivities towards perturbations.

Quadratic eigenvalue perturbation bounds; Generalized eigenvalue problem; Multiple eigenvalue

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