ON EXPLICIT RECURSIVE FORMULAS IN THE SPECTRAL PERTURBATION ANALYSIS OF A JORDAN BLOCK

作     者：Welters, Aaron 

作者机构：Univ Calif Irvine Dept Math Irvine CA 92697 USA 

出 版 物：《SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS》 (SIAM J. Matrix Anal. Appl.)

年 卷 期：2011年第32卷第1期

页      面：1-22页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Air Force Office of Scientific Research [FA9550-08-1-0103] 

主　　题：matrix perturbation theory degenerate eigenvalue Jordan block perturbation of eigenvalues and eigenvectors Puiseux series recursive formula 

摘      要：Let A(epsilon) be an analytic square matrix and lambda(0) an eigenvalue of A(0) of algebraic multiplicity m = 1. Then under the condition partial derivative/partial derivative epsilon det(lambda I - A(epsilon))|((epsilon,lambda)=(0,lambda 0)) not equal 0, we prove that the Jordan normal form of A(0) corresponding to the eigenvalue lambda(0) consists of a single m x m Jordan block, the perturbed eigenvalues near lambda(0) and their corresponding eigenvectors can be represented by a single convergent Puiseux series containing only powers of epsilon(1/m), and there are explicit recursive formulas to compute all the Puiseux series coefficients from just the derivatives of A(epsilon) at the origin. Using these recursive formulas we calculate the series coefficients up to the second order and list them for quick reference. This paper gives, under a generic condition, explicit recursive formulas to compute the perturbed eigenvalues and eigenvectors for non-self-adjoint analytic perturbations of matrices with nonderogatory eigenvalues.