Bounds For The Eigenvalues Of A Matrix
K. R. Garren · W&M Publish (College of William & Mary) · 1965
The purpose of this paper is to determine the eigenvalue bounds of a matrix defined over either the real or complex fields.Well known theorems concerning the condition of eigenvalues as a function of the condition of the related matrix are stated.Theorems which determine the bounds are derived.Closed form solutions are expressed in terms of (l) the matrix elements, (2) matrix norms, and (3) vectors and the eigenvalues of related matrices.A comparison is made in terms of the relative size of the areas of eigenvalue inclusion for the various solutions.Conditions for boundedness and unboundedness of these bounds are derived.Examples in terms of eigenvalue bounds for particular matrices are given.Results of this paper are used in the problem of determining critical points of a function of n variables and in the problem of determining the convergence of iterative solutions for a system of linear equations.v BOUNDS FOR THE EIGENVALUES OF A MATRIX For values of n H, the eigenvalues can always be found.How ever for n > r, this polynomial is not solvable by radicals.'*'Thus in general, for a matrix of order n > h, its eigenvalues cannot be found by direct means (in closed form solutions). Nevertheless, various techniques do exist for determining the bounds, both upper and lower, for the eigenvalues and quite often this information is sufficient to solve various types of problems.This paper will be concerned with theorems which will determine the upper and lower bounds for the-eigenvalues of a finite matrix defined over either the real or complex number fields.^B.L.