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Chapter 7: Q6E (page 395)

Find the matrix of the quadratic form. Assume x is in \({\mathbb{R}^{\bf{3}}}\).

a. \(3x_1^2 - 2x_2^2 + 5x_3^2 + 4{x_1}{x_2} - 6{x_1}{x_3}\)

b. \(4x_3^2 - 2{x_1}{x_2} + 4{x_2}{x_3}\)

Short Answer

Expert verified

a.\(\left( {\begin{aligned}{{}}3&2&{ - 3}\\2&{ - 2}&0\\{ - 3}&0&5\end{aligned}} \right)\)

b. \(\left( {\begin{aligned}{{}}0&{ - 1}&0\\{ - 1}&0&2\\0&2&4\end{aligned}} \right)\)

Step by step solution

01

Find an answer for part (a)

The coefficient matrix for the equation \(3{\bf{x}}_1^2 - 2{\bf{x}}_2^2 + 5{\bf{x}}_3^2 + 4{{\bf{x}}_1}{{\bf{x}}_2} - 6{{\bf{x}}_1}{{\bf{x}}_3}\).

\(\begin{aligned}{}Q\left( x \right) &= \left( {\begin{aligned}{{}}{{x_1}}&{{x_2}}&{{x_3}}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}3&{\frac{4}{2}}&{ - \frac{6}{2}}\\{\frac{4}{2}}&{ - 2}&{\frac{0}{2}}\\{ - \frac{6}{2}}&{\frac{0}{2}}&5\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}3&2&{ - 3}\\2&{ - 2}&0\\{ - 3}&0&5\end{aligned}} \right)\end{aligned}\)

The matrix in quadratic form is \(\left( {\begin{aligned}{{}}3&2&{ - 3}\\2&{ - 2}&0\\{ - 3}&0&5\end{aligned}} \right)\).

02

Find an answer for part (b)

The coefficient matrix for the equation \(4{\bf{x}}_3^2 - 2{{\bf{x}}_1}{{\bf{x}}_2} + 4{{\bf{x}}_2}{{\bf{x}}_3}\).

\(\begin{aligned}{}Q\left( x \right) &= \left( {\begin{aligned}{{}}{{x_1}}&{{x_2}}&{{x_3}}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}0&{ - \frac{2}{2}}&{ - \frac{0}{2}}\\{ - \frac{2}{2}}&0&{\frac{4}{2}}\\{\frac{0}{2}}&{\frac{4}{2}}&4\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}0&{ - 1}&0\\{ - 1}&0&2\\0&2&4\end{aligned}} \right)\end{aligned}\)

The matrix in quadratic form is \(\left( {\begin{aligned}{{}}0&{ - 1}&0\\{ - 1}&0&2\\0&2&4\end{aligned}} \right)\).

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Most popular questions from this chapter

Question: In Exercises 15 and 16, construct the pseudo-inverse of \(A\). Begin by using a matrix program to produce the SVD of \(A\), or, if that is not available, begin with an orthogonal diagonalization of \({A^T}A\). Use the pseudo-inverse to solve \(A{\rm{x}} = {\rm{b}}\), for \({\rm{b}} = \left( {6, - 1, - 4,6} \right)\) and let \(\mathop {\rm{x}}\limits^\^ \)be the solution. Make a calculation to verify that \(\mathop {\rm{x}}\limits^\^ \) is in Row \(A\). Find a nonzero vector \({\rm{u}}\) in Nul\(A\), and verify that \(\left\| {\mathop {\rm{x}}\limits^\^ } \right\| < \left\| {\mathop {\rm{x}}\limits^\^ + {\rm{u}}} \right\|\), which must be true by Exercise 13(c).

16. \(A = \left( {\begin{array}{*{20}{c}}4&0&{ - 1}&{ - 2}&0\\{ - 5}&0&3&5&0\\{\,\,\,2}&{\,\,0}&{ - 1}&{ - 2}&0\\{\,\,\,6}&{\,\,0}&{ - 3}&{ - 6}&0\end{array}} \right)\)

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.

17. \(\left( {\begin{aligned}{{}}1&{ - 6}&4\\{ - 6}&2&{ - 2}\\4&{ - 2}&{ - 3}\end{aligned}} \right)\)

In Exercises 17鈥�24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) , where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

24. Using the notation of Exercise 23, show that \({A^T}{u_j} = {\sigma _j}{v_j}\) for \({\bf{1}} \le {\bf{j}} \le {\bf{r}} = {\bf{rank}}\;{\bf{A}}\)

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 17鈥�24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) , where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

19. Show that the columns of\(V\)are eigenvectors of\({A^T}A\), the columns of\(U\)are eigenvectors of\(A{A^T}\), and the diagonal entries of\({\bf{\Sigma }}\)are the singular values of \(A\). (Hint: Use the SVD to compute \({A^T}A\) and \(A{A^T}\).)
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