13.1. SYMMETRIC AND ORTHOGONAL MATRICES 295
Example 13.1.16 Find an orthonormal set of three eigenvectors for the matrix
139
215
√5 8
45
√5
215
√5 6
5415
845
√5 4
156145
given the eigenvalues are 2, and 1.
The eigenvectors which go with λ = 2 are obtained from row reducing the matrix
139 −2 2
15
√5 8
45
√5 | 0
215
√5 6
5 −2 415 | 0
845
√5 4
156145 −2 | 0
and its row reduced echelon form is
1 0 − 12
√5 | 0
0 1 − 34 | 0
0 0 0 | 0
which shows the eigenvectors for λ = 2 are z
(12
√5 3
4 1)T
and a choice for z which
will produce a unit vector is z = 415
√5. Therefore, the vector we want is
(23
15
√5 4
15
√5)T
.
Next consider the eigenvectors for λ = 1. The matrix which must be row reduced is
139 −1 2
15
√5 8
45
√5 | 0
215
√5 6
5 −1 415 | 0
845
√5 4
156145 −1 | 0
and its row reduced echelon form is
1 310
√5 2
5
√5 | 0
0 0 0 | 00 0 0 | 0
.