368 CHAPTER 15. NUMERICAL METHODS, EIGENVALUES

Thus the eigenvalues are those diagonal entries. Now if you want to find eigenvectors, thereis a way to keep track of things and get them from the above, but you could also simply goback to the matrix A and use the shifted inverse power method. Starting with one of theseapproximate eigenvalues or a number close to one as α . I shall pick the eigenvalue 6.643and obtain an eigenvector and possibly a better approximation to this eigenvalue using theshifted inverse power method using the iterative procedure given above. This yields theeigenvector

u =(

0.6442 0.5961 1 0.5572)T

which works extremely well, along with the eigenvalue 6.643. In fact, the error betweenAu and 6.643u is on the order of 10−14.

15.7 Exercises1. Using the power method, find the eigenvalue correct to one decimal place having

largest absolute value for the matrix A =

 0 −4 −47 10 5−2 0 6

 along with an eigen-

vector associated with this eigenvalue.

2. Using the power method, find the eigenvalue correct to one decimal place having

largest absolute value for the matrix A =

 15 6 1−5 2 11 2 7

 along with an eigenvector

associated with this eigenvalue.

3. Using the power method, find the eigenvalue correct to one decimal place having

largest absolute value for the matrix A =

 10 4 2−3 2 −10 0 4

 along with an eigen-

vector associated with this eigenvalue.

4. Using the power method, find the eigenvalue correct to one decimal place having

largest absolute value for the matrix A =

 15 14 −3−13 −18 9

5 10 −1

 along with an

eigenvector associated with this eigenvalue.

5. In Example 15.4.3 an eigenvalue was found correct to several decimal places alongwith an eigenvector. Find the other eigenvalues along with their eigenvectors.

6. Find the eigenvalues and eigenvectors of the matrix A=

 3 2 12 1 31 3 2

 numerically.

In this case the exact eigenvalues are ±√

3,6. Compare with the exact answers.