214 CHAPTER 11. MATRICES AND THE INNER PRODUCT
Hint: Two eigenvalues are 18 and 24.
33. 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.
34. 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 =
− 5
31
15
√6√
5 815
√5
115
√6√
5 − 145 − 1
15
√6
815
√5 − 1
15
√6 7
15
Hint: The eigenvalues are −3,−2,1.
35. 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 =
3 0 00 3
212
0 12
32
.
36. 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 =
2 0 00 5 10 1 5
.
37. 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 =
43
13
√3√
2 13
√2
13
√3√
2 1 − 13
√3
13
√2 − 1
3
√3 5
3
Hint: The eigenvalues are 0,2,2 where 2 is listed twice because it is a root of multi-plicity 2.