Linear Algebra, Theory And Applications, 2012a

376 NUMERICAL METHODS FOR FINDING EIGENVALUES
15.1.2 The Explicit Description Of The Method
Here is how you use this method to find the eigenvalue and eigenvector closest
to α.
1. Find (A − αI) −1 .
2. Pick u 1 . If you are not phenomenally unlucky, the iterations will converge.
3. If u k has been obtained,
u k+1 = (A − αI)−1 u k
s k+1
where s k+1 is the entry of (A − αI) −1 u k which has largest absolute value.
4. When the scaling factors, s k are not changing much and the u k are not changing much,
find the approximation to the eigenvalue by solving
s k+1 = 1
λ − α
for λ. The eigenvector is approximated by u k+1 .
5. Check your work by multiplying by the original matrix to see how well what you have
found works.
Thus this amounts to the power method for the matrix (A − αI) −1 .
⎛
5 −14 11
⎞
Example 15.1.4 Find the eigenvalue of A = ⎝ −4 4 −4 ⎠ which is closest to −7.
3 6 −3
Also find an eigenvector which goes with this eigenvalue.
In this case the eigenvalues are −6, 0, and 12 so the correct answer is −6 for the eigenvalue.
Then from the above procedure, I will start with an initial vector,
⎛ ⎞
1
u 1 ≡ ⎝ 1 ⎠ .
1
Then I must solve the following equation.
⎛⎛
5 −14 11
⎞ ⎛
⎝⎝
−4 4 −4 ⎠ +7⎝
3 6 −3
1 0 0
0 1 0
0 0 1
Simplifying the matrix on the left, I must solve
⎛
⎞ ⎛
12 −14 11
⎝ −4 11 −4 ⎠ ⎝ x y
3 6 4 z
⎞⎞
⎛
⎠⎠
⎝
⎞ ⎛
⎠ =
x
y
z
⎝ 1 1
1
⎞ ⎛
⎠ = ⎝
and then divide by the entry which has largest absolute value to obtain
⎛
u 2 = ⎝ 1.0 ⎞
. 184 ⎠
−. 76
⎞
⎠
1
1
1
⎞
⎠