Chapter 7: Q30E (page 395)
Suppose Aand B are orthogonally diagonalizable and \(AB = BA\). Explain why \(AB\) is also orthogonally diagonalizable.
The matrix AB is orthogonally diagonizable.
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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 \({A^ + }A{A^ + } = {A^ + }\).
Show that if A is an \(n \times n\) symmetric matrix, then \(\left( {A{\bf{x}}} \right) \cdot {\bf{y}} = {\bf{x}} \cdot \left( {A{\bf{y}}} \right)\) for x, y in \({\mathbb{R}^n}\).
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.
13. \({\bf{ - }}x_{\bf{1}}^{\bf{2}}{\bf{ - 6}}{x_{\bf{1}}}{x_{\bf{2}}} + {\bf{9}}x_{\bf{2}}^{\bf{2}}\)
Determine which of the matrices in Exercises 7–12 are orthogonal. If orthogonal, find the inverse.
8. \(\left( {\begin{aligned}{{}}1&{\,\,\,1}\\1&{ - 1}\end{aligned}} \right)\)
Question: 6. Let A be an \(n \times n\) symmetric matrix. Use Exercise 5 and an eigenvector basis for \({\mathbb{R}^n}\) to give a second proof of the decomposition in Exercise 4(b).
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