Page 644 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Find the eigenvalues and bases for the eigenspaces of the linear operator defined by

Solution is

The matrix for T with respect to the standard basis

(verify). The eigenvalues of T are and (Example 5 of Section 7.1). Also from that example, the eigenspace of

corresponding to has the basis , where

and the eigenspace of corresponding to has the basis , where

The matrices , , and are the coordinate matrices relative to B of

Thus the eigenspace of T corresponding to has the basis

and that corresponding to has the basis

As a check, the reader should use the given formula for T to verify that , , and .

EXAMPLE 6 Diagonal Matrix for a Linear Operator
Let be the linear operator given by

Find a basis for relative to which the matrix for T is diagonal.

Solution

First we will find the standard matrix for T; then we will look for a change of basis that diagonalizes the standard matrix.
If denotes the standard basis for , then