328 CHAPTER 13. MATRICES AND THE INNER PRODUCT
30. 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.
31. 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
.
32. 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
.
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 =
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.
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 =
1 1
6
√3√
2 16
√3√
6
16
√3√
2 32
112
√2√
6
16
√3√
6 112
√2√
6 12