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Chapter 7: Q34E (page 395)

Construct a spectral decomposition of A from Example 3.

Short Answer

Expert verified

The spectral decomposition of matrixA is \(7\left[ {\begin{aligned}{{}}{\frac{1}{2}}&0&{\frac{1}{2}}\\0&0&0\\{\frac{1}{2}}&0&{\frac{1}{2}}\end{aligned}} \right] + 7\left[ {\begin{aligned}{{}}{\frac{1}{{18}}}&{ - \frac{4}{{18}}}&{ - \frac{1}{{18}}}\\{ - \frac{4}{{18}}}&{\frac{{16}}{{18}}}&{\frac{4}{{18}}}\\{ - \frac{1}{{18}}}&{\frac{4}{{18}}}&{\frac{1}{{18}}}\end{aligned}} \right] - 2\left[ {\begin{aligned}{{}}{\frac{4}{9}}&{\frac{2}{9}}&{ - \frac{4}{9}}\\{\frac{2}{9}}&{\frac{1}{9}}&{ - \frac{2}{9}}\\{ - \frac{4}{9}}&{ - \frac{2}{9}}&{\frac{4}{9}}\end{aligned}} \right]\).

Step by step solution

01

Write given values from example 3

The eigenvalues of the matrix \(A = \left[ {\begin{aligned}{{}}3&{ - 2}&4\\{ - 2}&6&2\\4&2&3\end{aligned}} \right]\)are 7, 7, and \( - 2\). The matrix P is defined as:

\(\begin{aligned}{}P &= \left[ {\begin{aligned}{{}}{{{\bf{u}}_1}}&{{{\bf{u}}_2}}&{{{\bf{u}}_3}}\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{{}}{\frac{1}{{\sqrt 2 }}}&{ - \frac{1}{{\sqrt {18} }}}&{ - \frac{2}{3}}\\0&{\frac{4}{{\sqrt {18} }}}&{ - \frac{1}{3}}\\{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt {18} }}}&{\frac{2}{3}}\end{aligned}} \right]\end{aligned}\)

02

Find the spectral decomposition of A

The spectral decomposition of A can be calculated as:

\(\begin{aligned}{}A &= {\lambda _1}{{\bf{u}}_1}{\bf{u}}_1^T + {\lambda _2}{{\bf{u}}_2}{\bf{u}}_2^T + {\lambda _3}{{\bf{u}}_3}{\bf{u}}_3^T\\ &= 8{{\bf{u}}_1}{\bf{u}}_1^T + 6{{\bf{u}}_2}{\bf{u}}_2^T + 6{{\bf{u}}_3}{\bf{u}}_3^T\\ &= 7\left[ {\begin{aligned}{{}}{ - \frac{1}{{\sqrt 2 }}}\\0\\{\frac{1}{{\sqrt 2 }}}\end{aligned}} \right]\left[ {\begin{aligned}{{}}{\frac{1}{{\sqrt 2 }}}&0&{\frac{1}{{\sqrt 2 }}}\end{aligned}} \right] + 6\left[ {\begin{aligned}{{}}{ - \frac{1}{{\sqrt {18} }}}\\{\frac{4}{{\sqrt {18} }}}\\{\frac{1}{{\sqrt {18} }}}\end{aligned}} \right]\left[ {\begin{aligned}{{}}{ - \frac{1}{{\sqrt {18} }}}&{\frac{4}{{\sqrt {18} }}}&{\frac{1}{{\sqrt {18} }}}\end{aligned}} \right] - 2\left[ {\begin{aligned}{{}}{ - \frac{2}{3}}\\{ - \frac{1}{3}}\\{\frac{2}{3}}\end{aligned}} \right]\left[ {\begin{aligned}{{}}{ - \frac{2}{3}}&{ - \frac{1}{3}}&{\frac{2}{3}}\end{aligned}} \right]\\ &= 7\left[ {\begin{aligned}{{}}{\frac{1}{2}}&0&{\frac{1}{2}}\\0&0&0\\{\frac{1}{2}}&0&{\frac{1}{2}}\end{aligned}} \right] + 7\left[ {\begin{aligned}{{}}{\frac{1}{{18}}}&{ - \frac{4}{{18}}}&{ - \frac{1}{{18}}}\\{ - \frac{4}{{18}}}&{\frac{{16}}{{18}}}&{\frac{4}{{18}}}\\{ - \frac{1}{{18}}}&{\frac{4}{{18}}}&{\frac{1}{{18}}}\end{aligned}} \right] - 2\left[ {\begin{aligned}{{}}{\frac{4}{9}}&{\frac{2}{9}}&{ - \frac{4}{9}}\\{\frac{2}{9}}&{\frac{1}{9}}&{ - \frac{2}{9}}\\{ - \frac{4}{9}}&{ - \frac{2}{9}}&{\frac{4}{9}}\end{aligned}} \right]\end{aligned}\)

Thus, the spectral matrix of A is \(7\left[ {\begin{aligned}{{}}{\frac{1}{2}}&0&{\frac{1}{2}}\\0&0&0\\{\frac{1}{2}}&0&{\frac{1}{2}}\end{aligned}} \right] + 7\left[ {\begin{aligned}{{}}{\frac{1}{{18}}}&{ - \frac{4}{{18}}}&{ - \frac{1}{{18}}}\\{ - \frac{4}{{18}}}&{\frac{{16}}{{18}}}&{\frac{4}{{18}}}\\{ - \frac{1}{{18}}}&{\frac{4}{{18}}}&{\frac{1}{{18}}}\end{aligned}} \right] - 2\left[ {\begin{aligned}{{}}{\frac{4}{9}}&{\frac{2}{9}}&{ - \frac{4}{9}}\\{\frac{2}{9}}&{\frac{1}{9}}&{ - \frac{2}{9}}\\{ - \frac{4}{9}}&{ - \frac{2}{9}}&{\frac{4}{9}}\end{aligned}} \right]\).

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Most popular questions from this chapter

Classify the quadratic forms in Exercises 9-18. Then make a change of variable, \({\bf{x}} = P{\bf{y}}\), that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. Construct P using the methods of Section 7.1.

10. \({\bf{2}}x_{\bf{1}}^{\bf{2}} + {\bf{6}}{x_{\bf{1}}}{x_{\bf{2}}} - {\bf{6}}x_{\bf{2}}^{\bf{2}}\)

In Exercises 25 and 26, mark each statement True or False. Justify each answer.

26.

  1. There are symmetric matrices that are not orthogonally diagonizable.
  2. b. If \(B = PD{P^T}\), where \({P^T} = {P^{ - {\bf{1}}}}\) and D is a diagonal matrix, then B is a symmetric matrix.
  3. c. An orthogonal matrix is orthogonally diagonizable.
  4. d. The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.

Let A be the matrix of the quadratic form

\({\bf{9}}x_{\bf{1}}^{\bf{2}} + {\bf{7}}x_{\bf{2}}^{\bf{2}} + {\bf{11}}x_{\bf{3}}^{\bf{2}} - {\bf{8}}{x_{\bf{1}}}{x_{\bf{2}}} + {\bf{8}}{x_{\bf{1}}}{x_{\bf{3}}}\)

It can be shown that the eigenvalues of A are 3,9, and 15. Find an orthogonal matrix P such that the change of variable \({\bf{x}} = P{\bf{y}}\) transforms \({{\bf{x}}^T}A{\bf{x}}\) into a quadratic form which no cross-product term. Give P and the new quadratic form.

Determine which of the matrices in Exercises 7鈥�12 are orthogonal. If orthogonal, find the inverse.

9. \(\left[ {\begin{aligned}{{}}{ - 4/5}&{\,\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right]\)

Orthogonally diagonalize the matrices in Exercises 13鈥�22, giving an orthogonal matrix\(P\)and a diagonal matrix\(D\). To save you time, the eigenvalues in Exercises 17鈥�22 are: (17)\( - {\bf{4}}\), 4, 7; (18)\( - {\bf{3}}\),\( - {\bf{6}}\), 9; (19)\( - {\bf{2}}\), 7; (20)\( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

13. \(\left( {\begin{aligned}{{}}3&1\\1&{\,\,3}\end{aligned}} \right)\)
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