298 CHAPTER 13. MATRICES AND THE INNER PRODUCT

Some matrices are non defective and some are not. As indicated in Theorem 13.1.13if A is a real symmetric matrix, there exists an orthogonal matrix U such that UT AU = Da diagonal matrix. Therefore, every symmetric matrix is non defective because if U is anorthogonal matrix, its inverse is UT . In the following example, this orthogonal matrix willbe found.

Example 13.1.19 Let A =

1 0 00 3

212

0 12

32

 . Find an orthogonal matrix U such that UT AU

is a diagonal matrix.

In this case, a tedious computation shows the eigenvalues are 2 and 1. First we will findan eigenvector for the eigenvalue 2. This involves row reducing the following augmentedmatrix. 

1 0 0 | 00 2− 3

2 − 12 | 0

0 − 12 2− 3

2 | 0

The row reduced echelon form is 1 0 0 | 0

0 1 −1 | 00 0 0 | 0

and so an eigenvector is

(0 1 1

)T. However, it is desired that the eigenvectors ob-

tained all be unit vectors and so dividing this vector by its length gives(0 1

2

√2 1

2

√2)T

.

Next consider the case of the eigenvalue, 1. The matrix which needs to be row reduced inthis case is 

0 0 0 | 00 1− 3

2 − 12 | 0

0 − 12 1− 3

2 | 0

The row reduced echelon form is 0 1 1 | 0

0 0 0 | 00 0 0 | 0

 .

Therefore, the eigenvectors are of the form(

s −t t)T

. Two of these which are or-thonormal are  1

00

 and

 0−1/√

21/√

2

 .