Page 537 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

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CHAPTER

Eigenvalues, Eigenvectors

I N T R O D U C T I O N : If A is an n x n matrix and x is a vector in Rn , then Ax is also a vector in Rn, but usually there is no

simple geometric relationship between x and . However, in the special case where x is a nonzero vector and is a scalar

multiple of x, a simple geometric relationship occurs. For example, if A is a matrix, and if x is a nonzero vector such that

is a scalar multiple of x, say , then each vector on the line through the origin determined by x gets mapped back

onto the same line under multiplication by

Nonzero vectors that get mapped into scalar multiples of themselves under a linear operator arise naturally in the study of
vibrations, genetics, population dynamics, quantum mechanics, and economics, as well as in geometry. In this chapter we will
study such vectors and their applications.

Copyright © 2005 John Wiley & Sons, Inc. All rights reserved.