Numerical Linear Algebra with Applications

Chapter 18

The Algebraic Eigenvalue Problem

Abstract

The chapter begins with three applications to show the importance of the eigenvalue problem. The problems deal with resonance, the Leslie matrix, and column buckling. The power method is the first eigenvalue solver to be discussed. It computes the largest eigenvalue in magnitude as long as it is simple. The inverse power iteration computes the eigenvalue of smallest magnitude by computing the largest eigenvalue of the inverse. Both the methods actually compute the eigenvector associated with the desired eigenvalue, and then the Rayleigh quotient finds the eigenvalue. The basic QR iteration is the foundation for most eigenvalue solvers. By itself, it is not very useful, but when applied ...