15.7. EXERCISES 369

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

 3 2 12 5 31 3 2

 numerically.

The exact eigenvalues are 2,4+√

15,4−√

15. Compare your numerical results withthe exact values. Is it much fun to compute the exact eigenvectors?

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

 0 2 12 5 31 3 2

 numerically.

We don’t know the exact eigenvalues in this case. Check your answers by multiplyingyour numerically computed eigenvectors by the matrix.

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

 0 2 12 0 31 3 2

 numerically.

We don’t know the exact eigenvalues in this case. Check your answers by multiplyingyour numerically computed eigenvectors by the matrix.

10. Consider the matrix A =

 3 2 32 1 43 4 0

 and the vector (1,1,1)T . Estimate the dis-

tance between the Rayleigh quotient determined by this vector and some eigenvalueof A.

11. Consider the matrix A =

 1 2 12 1 41 4 5

 and the vector (1,1,1)T . Estimate the dis-

tance between the Rayleigh quotient determined by this vector and some eigenvalueof A.

12. Using Gerschgorin’s theorem, find upper and lower bounds for the eigenvalues of

A =

 3 2 32 6 43 4 −3

 .

13. The QR algorithm works very well on general matrices. Try the QR algorithm on thefollowing matrix which happens to have some complex eigenvalues.

A =

 1 2 31 2 −1−1 −1 1

Use the QR algorithm to get approximate eigenvalues and then use the shifted in-verse power method on one of these to get an approximate eigenvector for one of thecomplex eigenvalues.