The Eigenvalue Shift Technique and Its Eigenstructure Analysis of a Matrix

TL;DR

This paper introduces a new method for updating multiple eigenvalues of a matrix and analyzes the resulting changes in its Jordan canonical form, enhancing the eigenvalue shift technique's applicability.

Contribution

It proposes a novel approach for shifting eigenvalues with multiplicities and thoroughly examines the resulting eigenstructure changes.

Findings

New eigenvalue update method for multiple eigenvalues

Analysis of Jordan canonical form after eigenvalue shift

Enhanced understanding of eigenstructure modifications

Abstract

The eigenvalue shift technique is the most well-known and fundamental tool for matrix computations. Applications include the search of eigeninformation, the acceleration of numerical algorithms, the study of Google's PageRank. The shift strategy arises from the concept investigated by Brauer [1] for changing the value of an eigenvalue of a matrix to the desired one, while keeping the remaining eigenvalues and the original eigenvectors unchanged. The idea of shifting distinct eigenvalues can easily be generalized by Brauer's idea. However, shifting an eigenvalue with multiple multiplicities is a challenge issue and worthy of our investigation. In this work, we propose a new way for updating an eigenvalue with multiple multiplicities and thoroughly analyze its corresponding Jordan canonical form after the update procedure.