Chapter 7: Q5E (page 395)
Determine which of the matrices in Exercises 1–6 are symmetric.
5. \(\left( {\begin{aligned}{{}{}}{ - 6}&2&0\\2&{ - 6}&2\\0&2&{ - 6}\end{aligned}} \right)\)
The given matrix is symmetric.
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Question: In Exercises 17-22, determine which sets of vectors are orthonormal. If a set is only orthogonal, normalize the vectors to produce an orthonormal set.
18. \(\left( {\begin{array}{*{20}{c}}0\\1\\0\end{array}} \right),{\rm{ }}\left( {\begin{array}{*{20}{c}}0\\{ - 1}\\0\end{array}} \right)\)
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.
14. \({\bf{3}}x_{\bf{1}}^{\bf{2}} + {\bf{4}}{x_{\bf{1}}}{x_{\bf{2}}}\)
Orhtogonally diagonalize the matrices in Exercises 37-40. To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and for each eigenvalue \(\lambda \), find an orthogonal basis for \({\bf{Nul}}\left( {A - \lambda I} \right)\), as in Examples 2 and 3.
37. \(\left( {\begin{aligned}{{}}{\bf{6}}&{\bf{2}}&{\bf{9}}&{ - {\bf{6}}}\\{\bf{2}}&{\bf{6}}&{ - {\bf{6}}}&{\bf{9}}\\{\bf{9}}&{ - {\bf{6}}}&{\bf{6}}&{\bf{2}}\\{\bf{6}}&{\bf{9}}&{\bf{2}}&{\bf{6}}\end{aligned}} \right)\)
All symmetric matrices are diagonalizable.
Question: 12. Exercises 12–14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).
Verify the properties of\({A^ + }\):
a. For each\({\rm{y}}\)in\({\mathbb{R}^m}\),\(A{A^ + }{\rm{y}}\)is the orthogonal projection of\({\rm{y}}\)onto\({\rm{Col}}\,A\).
b. For each\({\rm{x}}\)in\({\mathbb{R}^n}\),\({A^ + }A{\rm{x}}\)is the orthogonal projection of\({\rm{x}}\)onto\({\rm{Row}}\,A\).
c. \(A{A^ + }A = A\)and \({A^ + }A{A^ + } = {A^ + }\).
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