11.8. EXERCISES 215
38. 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
Hint: The eigenvalues are 2,1,0.
39. Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
13
16
√3√
2 − 718
√3√
6
16
√3√
2 32 − 1
12
√2√
6
− 718
√3√
6 − 112
√2√
6 − 56
Hint: The eigenvalues are 1,2,−2.
40. Find the eigenvalues and an orthonormal basis of eigenvectors for the matrix
− 1
2 − 15
√6√
5 110
√5
− 15
√6√
5 75 − 1
5
√6
110
√5 − 1
5
√6 − 9
10
Hint: The eigenvalues are −1,2,−1 where −1 is listed twice because it has multi-plicity 2 as a zero of the characteristic equation.
41. 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.
42. You are doing experiments and have obtained the ordered pairs,
(0,1) ,(1,2) ,(2,3.5) ,(3,4)
Find m and b such that y = mx+b approximates these four points as well as possible.Now do the same thing for y = ax2 + bx + c, finding a,b, and c to give the bestapproximation.
43. Suppose you have several ordered triples, (xi,yi,zi) . Describe how to find a polyno-mial,
z = a+bx+ cy+dxy+ ex2 + f y2