The Condition Numbers of Semi-simple Eigenvalue of Quadratic Eigenvalue Problem

Abstract

The condition numbers of eigenvalues of matrices measure the sensitivities of eigenvalues to small perturbation of matrix. They're widely used to assess the quality of numerical algorithms for eigenvalue problems. This paper considers the condition number of multiple eigenvalue of regular quadratic eigenvalue problem. Based on the properties of multiple eigenvalue of quadratic eigenvalue problem analytically dependent on several parameters, we give various definitions for condition numbers of semi-simple eigenvalue of regular quadratic eigenvalue problem. Utilizing SVD and the properties of unitarily invariant norm, we derive the computational expressions for the introduced condition numbers. We find that the condition numbers defined can be computed in terms of the singular values of , where  are respectively the right eigenvector matrix and left eigenvector matrix corresponding to the multiple eigenvalue. Compared with the existing condition numbers of multiple eigenvalues of quadratic eigenvalue problem, the condition numbers defined in this paper can measure not only the worst case sensitivity of semi-simple eigenvalue, but also the different sensitivities of the eigenvalues spawned from semi-simple eigenvalue.