A First Course in Linear Algebra, 2017a

7.4. Orthogonality 429
⎡
A =
⎢
⎣
√ √
4 1
3 3√
3 2
1
⎤
3
2
√ √ 1
3√
3 2 1 −
1 3 3
√ √
⎥
1
3 2 −
1
3
3
5 ⎦
3
Hint: The eigenvalues are 0,2,2 where 2 is listed twice because it is a root of multiplicity 2.
Exercise 7.4.10 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 U T AU = D.
Hint: The eigenvalues are 2,1,0.
⎡
√ √ √ √
1
1 6 3 2
1
⎤
6
3 6
√ √ √ 1
A =
6√
3 2
3 1
2 12
2 6
⎢
⎣ √ √ √ √
⎥
1
6 3 6
1
12
2 6
1 ⎦
2
Exercise 7.4.11 Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
⎡
A =
⎢
⎣
Hint: The eigenvalues are 1,2,−2.
1
6
− 7
18
√ √ √ √ ⎤
1 1
3 6 3 2 −
7
18
3 6
√ √
3 2
3
2
−
12√ 1 √ 2 6
√ √ √ √ ⎥
3 6 −
1
12
2 6 −
5 ⎦
6
Exercise 7.4.12 Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
⎡
− 1 2
− 1 √ √
5√
6 5
1
10
5
A =
− 1 √ 5√
6 5
7
5
− 1 5√
6
⎢
⎣ √ √ ⎥
5 −
1
5
6 −
9 ⎦
1
10
10
⎤
Hint: The eigenvalues are −1,2,−1 where −1 is listed twice because it has multiplicity 2 as a zero of
the characteristic equation.
Exercise 7.4.13 Explain why a matrix A is symmetric if and only if there exists an orthogonal matrix U
such that A = U T DU for D a diagonal matrix.
Exercise 7.4.14 Show that if A is a real symmetric matrix and λ and μ are two different eigenvalues, then
if X is an eigenvector for λ and Y is an eigenvector for μ, then X •Y = 0. Also all eigenvalues are real.