19.9. EXERCISES 429

27. 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 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.

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 =

3 0 00 3

212

0 12

32

 .

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 =

 2 0 00 5 10 1 5

 .

32. Explain why a real matrix A is symmetric if and only if there exists an orthogonalmatrix U such that A =UT DU for D a diagonal matrix.

33. Find an orthonormal basis for the spans of the following sets of vectors.

(a) (3,−4,0) ,(7,−1,0) ,(1,7,1).

(b) (3,0,−4) ,(11,0,2) ,(1,1,7)

(c) (3,0,−4) ,(5,0,10) ,(−7,1,1)