Improving the condition number of a simple eigenvalue by a rank one matrix

Abstract

In this work a technique to improve the condition number si of a simple eigenvalue lambda(i) of a matrix A is an element of C-nxn is given. This technique obtains a rank one updated matrix that is similar to A with the eigenvalue condition number of lambda(i) equal to one. More precisely, the similar updated matrix A + v(i)q*, where Av(i) = lambda(i)v(i) and q is a fixed vector, has s(i) = 1 and the remaining condition numbers are at most equal to the corresponding initial condition numbers. Moreover an expression to compute the vector q, using only the eigenvalue lambda(i) and its eigenvector v(i), is given. (C) 2016 Elsevier Ltd. All rights reserved.