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

Autores UPV
Año
Revista APPLIED MATHEMATICS LETTERS

Abstract

In this work a technique to improve the condition number $s_i$ of a simple eigenvalue $\lambda_i$ of a matrix $A\in \mathbb{C}^{n \times n}$ 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 $A+v_iq^{\ast}$, where $Av_i=\lambda_iv_i$ 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.