The Research On The Perturbation Bounds Of Eigenvalues For The Hermitian Matrix

Matrix eigenvalue perturbation boundary problem is the change scope of eigenvalues when the matrix is perturbed, the degree of change is called the perturbation bounds of eigenvalue. Eigenvalue has the relationship with the matrix and the corresponding eigenvectors, therefore, studying the perturbed matrix and the perturbation matrix’ structure characteristic, and the change of the eigenvector, is an important approach to study the eigenvalue perturbation bounds. It is very important to focuse on the study of Hermitian matrix eigenvalue perturbation bounds, analyze the special matrix perturbation, and give the optimization of the eigenvalue perturbation bounds for Hermitian matrix.In this paper, we mainly talked about perturbation of partitioned Hermitian eigenvalue problem, the perturbation of multiple eigenvalues and perturbation bounds for generalized Hermitian matrix eigenvalues. The paper optimizes partitioned Hermitian eigenvalue perturbation bounds, popularized the multiple eigenvalues of Hermitian matrix and the multiple eigenvalues of generalized Hermitian matrix.This paper mainly studies the following contents.For the sharp perturbation bound of 2-by-2 block Hermitian matrix, we define the new perturbation bounds for the generalized Hermitian matrix multiple eigenvalues basing on author Y. J. Nakatsukasa. We mainly sharp the eigenvalue bounds through bounding the more accurate eigenvector, finding that the bounds of eigenvector are relactive to the spectral norm of component of 2-by-2 perturbation matrix, and then analysis the single eigenvalue’s sensitivity to perturbation depend on the eigenvector of the corresponding components of bounded size, that is associated with the 2-by-2 block perturbation matrix component.For the standard eigenvalue perturbation problem, we mainly generalize perturbation bounds of multiple eigenvalues, improve the corresponding Hermitian matrix eigenvalue perturbation error bound, obtain the more accurate perturbation bound.For the generalized eigenvalue problem of symmetric perturbation, we give a new form of a matrix, improve the corresponding Hermitian matrix generalized eigenvalue perturbation error bound, in order to get more general perturbation bounds.