13.9. EXERCISES 327
24. Fill in the missing entries to make the matrix orthogonal.−1√
2−1√
61√3
1√2
√6
3
.
25. Fill in the missing entries to make the matrix orthogonal.23
√2
216
√2
23
0
26. Fill in the missing entries to make the matrix orthogonal.
13 − 2√
523 0
415
√5
27. Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize A
by finding an orthogonal matrix U and a diagonal matrix D such that UT AU = D.
A =
−1 1 11 −1 11 1 −1
.
Hint: One eigenvalue is -2.
28. Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize Aby finding an orthogonal matrix U and a diagonal matrix D such that UT AU = D.
A =
17 −7 −4−7 17 −4−4 −4 14
.
Hint: Two eigenvalues are 18 and 24.
29. Find the eigenvalues and an orthonormal basis of eigenvectors for A. Diagonalize Aby finding an orthogonal matrix U and a diagonal matrix D such that UT AU = D.
A =
13 1 41 13 44 4 10
.
Hint: Two eigenvalues are 12 and 18.