Page 160 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

This system can be rewritten as
or
or
which is of form 7 with

The primary problem of interest for linear systems of the form 7 is to determine those values of for which the system has a
nontrivial solution; such a value of is called a characteristic value or an eigenvalue* of . If is an eigenvalue of , then
the nontrivial solutions of 7 are called the eigenvectors of A corresponding to .

It follows from Theorem 2.3.3 that the system has a nontrivial solution if and only if

(8)

This is called the characteristic equation of ; the eigenvalues of A can be found by solving this equation for .

Eigenvalues and eigenvectors will be studied again in subsequent chapters, where we will discuss their geometric
interpretation and develop their properties in more depth.

EXAMPLE 6 Eigenvalues and Eigenvectors
Find the eigenvalues and corresponding eigenvectors of the matrix A in Example 5.

Solution

The characteristic equation of A is

The factored form of this equation is , so the eigenvalues of A are and .
By definition,

is an eigenvector of A if and only if is a nontrivial solution of ; that is,

(9)