A First Course in Linear Algebra, 2017a

402 Spectral Theory
However, it is desired that the eigenvectors be unit vectors and so dividing this vector by its length gives
⎡
⎢
⎣
⎤
0
1√ 2 ⎥
⎦
1√
2
Next find the eigenvectors corresponding to the eigenvalue equal to 1. The appropriate augmented matrix
and resulting reduced row-echelon form are given by:
⎡
⎢
⎣
1 − 1 0 0 0
0 1− 3 2
− 1 2
0
0 − 1 2
1 − 3 2
0
Therefore, the eigenvectors are of the form
Two of these which are orthonormal are ⎣
⎡
1
0
0
⎤
⎡
⎣
⎤
⎡
⎥
⎦ →···→ ⎣
s
−t
t
⎤
⎦
0 1 1 0
0 0 0 0
0 0 0 0
⎦, choosing s = 1andt = 0, and
⎤
⎦
⎡
⎢
⎣
0
− 1 √
2
1√
2
⎤
⎥
⎦ , letting s = 0,
t = 1 and normalizing the resulting vector.
To obtain the desired orthogonal matrix, we let the orthonormal eigenvectors computed above be the
columns.
⎡
⎤
0 1 0
⎢
⎣ − √ 1 1
0 √2 ⎥
2 ⎦
1√ 1
2
0 √2
To verify, compute U T AU as follows:
⎡
0 − √ 1 √2 1 2
U T AU = ⎢ 1 0 0
⎣
0
1 √2
⎤
⎥
√2 1 ⎦
⎡
⎢
⎣
1 0 0
0 3 2
0 1 2
1
2
3
2
⎤⎡
⎥⎢
⎦⎣
⎤
0 1 0
− √ 1 1
0 √2 ⎥
2 ⎦
1√ 1
0 √2
2
⎡
= ⎣
1 0 0
0 1 0
0 0 2
⎤
⎦ = D
the desired diagonal matrix. Notice that the eigenvectors, which construct the columns of U, are in the
same order as the eigenvalues in D.
♠
We conclude this section with a Theorem that generalizes earlier results.