Perturbation Theorems for Matrix Eigenvalues

概要

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The matrix eigenvalue problem arises in a wide variety of areas in the physical and social sciences as well as in engineering, most typically, for example, in the stability analysis of physical systems that are modeled by linear systems of equations, differential equations, and so on. Perturbation theorems on matrix eigenvalues are concerned with localization of eigenvalues, i.e., to produce regions in the complex plane in which eigenvalues of a given matrix lie. The theorems place bounds on the variation of the eigenvalues in terms of the variation of matrix elements. The information given by the theorems is useful in estimating true eigenvalues from computed or approximate eigenvalues, in analysing the stability of eigenvalues, and so on. In this paper we are concerned with a unified derivation of a class of common perturbation theorems for matrix eigenvalues. To this end we prove first a basic inequality (see (2.1) below) which appears to be unreported in the literature. Some of the inequalities presented in this paper are well-known while others such as (2.1) and (3.9) appear to be less well-known despite their usefulness.
一般社団法人情報処理学会の論文
1983-07-20