A First Course in Linear Algebra, 2017a

400 Spectral Theory
Solution. You can verify that the eigenvalues of A are 9 (with multiplicity two) and 18 (with multiplicity
one). Consider the eigenvectors corresponding to λ = 9. The appropriate augmented matrix and reduced
row-echelon form are given by
and so eigenvectors are of the form
⎡
⎣
9 − 10 −2 −2 0
−2 9− 13 −4 0
−2 −4 9− 13 0
⎡
⎣
⎤
−2y − 2z
y
z
⎡
⎦ →···→⎣
⎤
⎦
1 2 2 0
0 0 0 0
0 0 0 0
⎡We need ⎤ to find two of these which are orthogonal. Let one be given by setting z = 0andy = 1, giving
−2
⎣ 1 ⎦.
0
In order to find an eigenvector orthogonal to this one, we need to satisfy
⎡ ⎤
−2
⎡ ⎤
−2y − 2z
⎣ 1 ⎦ • ⎣
0
y
z
⎦ = 5y + 4z = 0
The values y = −4 andz = 5 satisfy this equation, giving another eigenvector corresponding to λ = 9as
⎡
⎤
−2(−4) − 2(5)
⎡ ⎤
−2
⎣ (−4)
5
⎦ = ⎣ −4 ⎦
5
Next find the eigenvector for λ = 18. The augmented matrix and the resulting reduced row-echelon
form are given by
⎡
⎤ ⎡
18 − 10 −2 −2 0
1 0 − 1 ⎤
2
0
⎣ −2 18− 13 −4 0 ⎦ →···→⎣
0 1 −1 0 ⎦
−2 −4 18− 13 0
0 0 0 0
⎤
⎦
and so an eigenvector is
⎡
⎣
1
2
2
⎤
⎦
Dividing each eigenvector by its length, the orthonormal set is
⎧ ⎡ ⎤
⎨ −2 √
⎡ ⎤ ⎡
−2
1
√ ⎣ 5
1 ⎦, ⎣ −4 ⎦, 1 ⎣
⎩ 5 15 3
0
5
1
2
2
⎤⎫
⎬
⎦
⎭
♠