Chapter 7: Q28E (page 395)
Suppose A is a symmetric \(n \times n\) matrix and B is any \(n \times m\) matrix. Show that \({B^T}AB\), \({B^T}B\), and \(B{B^T}\) are symmetric matrices.
The matrices \({B^T}AB\), \({B^T}B\), and \(B{B^T}\) are symmetric matrices.
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Suppose\(A = PR{P^{ - {\bf{1}}}}\), where P is orthogonal and R is upper triangular. Show that if A is symmetric, then R is symmetric and hence is actually a diagonal matrix.
Determine which of the matrices in Exercises 7–12 are orthogonal. If orthogonal, find the inverse.
7. \(\left( {\begin{aligned}{{}{}}{.6}&{\,\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\)
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.
15. \( - {\bf{3}}x_{\bf{1}}^{\bf{2}} - {\bf{7}}x_{\bf{2}}^{\bf{2}} - {\bf{10}}x_{\bf{3}}^{\bf{2}} - {\bf{10}}x_{\bf{4}}^{\bf{2}} + {\bf{4}}{x_{\bf{1}}}{x_{\bf{2}}} + {\bf{4}}{x_{\bf{1}}}{x_{\bf{3}}} + {x_{\bf{1}}}{x_{\bf{4}}} + {\bf{6}}{x_{\bf{3}}}{x_{\bf{4}}}\)
Question: If A is \(m \times n\), then the matrix \(G = {A^T}A\) is called the Gram matrix of A. In this case, the entries of G are the inner products of the columns of A. (See Exercises 9 and 10).
9. Show that the Gram matrix of any matrix A is positive semidefinite, with the same rank as A. (See the Exercises in Section 6.5.)
Question: 2. Let \(\left\{ {{{\bf{u}}_1},{{\bf{u}}_2},....,{{\bf{u}}_n}} \right\}\) be an orthonormal basis for \({\mathbb{R}_n}\) , and let \({\lambda _1},....{\lambda _n}\) be any real scalars. Define
\(A = {\lambda _1}{{\bf{u}}_1}{\bf{u}}_1^T + ..... + {\lambda _n}{\bf{u}}_n^T\)
a. Show that A is symmetric.
b. Show that \({\lambda _1},....{\lambda _n}\) are the eigenvalues of A
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