Page 645 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

so the standard matrix for T is

(13)

We now want to change from the standard basis B to a new basis in order to obtain a diagonal matrix for T. If

we let P be the transition matrix from the unknown basis to the standard basis B, then by Theorem 8.5.2, the matrices

and will be related by

(14)

In Example 1 of Section 7.2, we found that the matrix in 13 is diagonalized by

Since P represents the transition matrix from the basis to the standard basis , the columns of
P are , , and , so

Thus

are basis vectors that produce a diagonal matrix for .

As a check, let us compute directly. From the given formula for T, we have

so that

Thus

This is consistent with 14 since