Easy Algorithms for Finding Eigenvalues

Abstract

Many students in an introductory linear algebra course realize that writing down the characteristic equation for a modest size matrix would be an algebraic nightmare. Even if the characteristic equation is known, finding its roots may seem hard. This leads many students to believe that it is not practical to try to compute the eigenvalues for large or modest size matrices. A brief introduction to appropriate numerical techniques can dispel that conviction. The simplest numerical technique for estimating eigenvectors is the power method. Unfortunately convergence of the power method can be quite slow. However, a second simple technique, the Rayleigh quotient iteration, converges at a dazzling rate. We will see an example where we are able to estimate accurately an eigenvalue of a 3 by 3 matrix to 16 significant figures by solving four linear systems of equations and by using a little vector arithmetic. Both the power method and the Rayleigh quotient iteration are easy to experiment with in an interactive matrix arithmetic environment on a computer. This note collects the necessary results so that students will have an easy time running such experiments on a computer. We will be able to compare the power method and the Rayleigh quotient iteration with a posteriori error estimates. All of this is a nice glimpse of numerical linear algebra. Moreover, students get a chance to experiment with algorithms for finding eigenvalues that converge quickly and will generalize to large matrices with little difficulty.