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Short cuts to computing eigenvalues.

Article Excerpt
Abstract

We suggest some short cuts to compute eigenvalues be incorporated into standard elementary linear algebra texts. These shortcuts, whose proofs are quite elementary, have been classroom-tested and favored by our students over the usual methods. In addition to saving time and effort, these short cuts also underline and shed some light on the relationships that exist between several important concepts in linear algebra such as trace, determinant, block matrices, and eigenvalues.

Introduction

Whether one needs to find the stable state of the populations of coyotes and road runners living in Arizona, or to predict the long term winning record of a baseball team under certain assumptions based on its past record, or even to classify a quadratic form axsup(2) + 2bxy + cysup(2) + dx + ey + f = as a parabola, hyperbola, ellipse, or a circle (we use xsup(n) to denote x raised to the power n), as well as to solve a host of other problems ranging from physics and economics to differential equations and dynamical systems, one customarily use techniques from linear algebra that depend on computing the eigenvalues of some associated matrices [4], Chapter 9. The eigenvalues of an n by n matrix A are the zeros of the characteristic polynomial of A. This is a polynomial f(x) of degree n that is equal to the determinant of the matrix A--xI(n) where I(n) represents the identity matrix of order n. Thus, finding the eigenvalues of an n by n matrix A involves the computation of the zeros of the characteristic polynomial of the matrix. For n = 2, the computation does not present any difficulty as one can use the quadratic formula. However, if the degree of the characteristic polynomial is larger than 2, the problem...

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