11.8. EXERCISES 213
25. Suppose A is a 3×3 symmetric matrix and you have found two eigenvectors whichform an orthonormal set. Explain why their cross product is also an eigenvector.
26. Determine which of the following sets of vectors are orthonormal sets. Justify youranswer.
(a) {(1,1) ,(1,−1)}
(b){(
1√2, −1√
2
),(1,0)
}(c)
{( 13 ,
23 ,
23
),(−2
3 , −13 , 2
3
),( 2
3 ,−23 , 1
3
)}27. Show that if {u1, · · · ,un} is an orthonormal set of vectors in Fn, then it is a basis.
Hint: It was shown earlier that this is a linearly independent set.
28. Fill in the missing entries to make the matrix orthogonal.−1√
2−1√
61√3
1√2
√6
3
.
29. Fill in the missing entries to make the matrix orthogonal.23
√2
216
√2
23
0
30. Fill in the missing entries to make the matrix orthogonal.
13 − 2√
523 0
415
√5
31. 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.
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 =
17 −7 −4−7 17 −4−4 −4 14
.