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9.2 Diagonalization 277

SECTION 9.1 PROBLEMS

⎛ ⎞
In each of Problems 1 through 16, find the eigenvalues −2 1 0 0
of the matrix. For each eigenvalue, find an eigenvector. ⎜ 1 0 0 1 ⎟
14. ⎜ ⎟
Sketch the Gershgorin circles for the matrix and locate the ⎝ 0 0 0 0 ⎠
eigenvalues as points in the plane. 0 0 0 0
⎛ ⎞
−4 1 0 1
1 3
1. 0 1 0 0
2 1 15. ⎜ ⎟
⎜
⎟
⎝ 0 0 2
0 ⎠
−2 0
2. 1 0 0 3
1 4
⎛ ⎞
5 1 0 9

−5 0
3. 16. ⎜ 0 1 0 9 ⎟
1 2 ⎜ 0 0 9 ⎠
⎟
⎝ 0
0 0 0 0
6 −2
4.
−3 4
In each of Problems 17 through 22, find the eigenval-

1 −6
5. ues and associated eigenvectors of the matrix. Verify
2 2
that eigenvectors associated with distinct eigenvalues are

0 1 orthogonal.
6.
0 0

⎛ ⎞ 4 −2
2 0 0 17.
7. ⎝ 1 0 2 ⎠ −2 1
0 0 3 −3 5
18.
⎛ ⎞ 5 4
−2 1 0
8. ⎝ 1 3 0 ⎠ 6 1
0 0 −1 19. 1 4
−3 1 1 −13 1
⎛ ⎞
9. ⎝ 0 0 0 ⎠ 20. 1 4
0 1 0 ⎛ ⎞
0 1 0
⎛ ⎞
0 0 −1 21. ⎝ 1 −2 0 ⎠
10. ⎝ 0 0 1 ⎠ 0 0 3
2 0 0 ⎛ ⎞
0 1 1
⎛ ⎞
−14 1 0 22. ⎝ 1 2 0 ⎠
11. ⎝ 0 2 0 ⎠ 1 0 2
1 0 2
23. Suppose λ is an eigenvalue of A with eigenvector
⎛ ⎞ k
3 0 0 E.Let k be a positive integer. Show that λ is an
k
12. ⎝ 1 −2 −8 ⎠ eigenvalue of A with eigenvector E.
0 −5 1 24. Let A be an n × n matrix of numbers. Show that the
⎛ ⎞ constant term in the characteristic polynomial of A is
1 −2 0
n
13. ⎝ 0 0 0 ⎠ (−1) |A|. Use this to show that any singular matrix
−5 0 7 must have 0 as an eigenvalue.
9.2 Diagonalization
Recall that the elements a ii of a matrix make up its main diagonal. All other matrix elements are
called off-diagonal elements.
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