Linear Algebra, Theory And Applications, 2012a

158 SPECTRAL THEORY
As an example, consider the following.
Example 7.1.3 Find the eigenvalues and eigenvectors for the matrix
⎛
5 −10
⎞
−5
A = ⎝ 2 14 2 ⎠ .
−4 −8 6
You first need to identify the eigenvalues. Recall this requires the solution of the equation
⎛ ⎛
det ⎝λ ⎝ 1 0 0
⎞ ⎛
⎞⎞
5 −10 −5
0 1 0 ⎠ − ⎝ 2 14 2 ⎠⎠ =0
0 0 1 −4 −8 6
When you expand this determinant, you find the equation is
(λ − 5) ( λ 2 − 20λ + 100 ) =0
and so the eigenvalues are
5, 10, 10.
I have listed 10 twice because it is a zero of multiplicity two due to
λ 2 − 20λ + 100 = (λ − 10) 2 .
Having found the eigenvalues, it only remains to find the eigenvectors. First find the
eigenvectors for λ =5. As explained above, this requires you to solve the equation,
⎛ ⎛
⎝5 ⎝ 1 0 0 ⎞ ⎛
⎞⎞
⎛
5 −10 −5
0 1 0 ⎠ − ⎝ 2 14 2 ⎠⎠
⎝ x ⎞ ⎛
y ⎠ = ⎝ 0 ⎞
0 ⎠ .
0 0 1 −4 −8 6 z 0
That is you need to find the solution to
⎛
⎝ −2 −9 −2
0 10 5
4 8 −1
⎞ ⎛
⎠ ⎝ x ⎞ ⎛
y ⎠ = ⎝ 0 ⎞
0 ⎠
z 0
By now this is an old problem. You set up the augmented matrix and row reduce to get the
solution. Thus the matrix you must row reduce is
⎛
⎞
−2 −9 −2 0 ⎠ . (7.3)
⎝ 0 10 5 0
4 8 −1 0
The reduced row echelon form is
⎛
⎞
1 0 − 5 4
0
⎝
1
0 1
2
0 ⎠
0 0 0 0
and so the solution is any vector of the form
⎛
5
4
⎝
z
⎞ ⎛
−1
2 z ⎠ = z ⎝
z
5
4
−1
2
1
⎞
⎠