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Chapter 7: Q27E (page 395)
Let \(A\) and \(B\)be symmetric \(n \times n\) matrices whose eigenvalues are all positive. Show that the eigenvalues of\(A + B\) are all positive.
It is proved that the eigenvalues of \(A + B\) are all positive.
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Determine which of the matrices in Exercises 1–6 are symmetric.
5. \(\left( {\begin{aligned}{{}{}}{ - 6}&2&0\\2&{ - 6}&2\\0&2&{ - 6}\end{aligned}} \right)\)
Let \(A = \left( {\begin{aligned}{{}}{\,\,\,4}&{ - 1}&{ - 1}\\{ - 1}&{\,\,\,4}&{ - 1}\\{ - 1}&{ - 1}&{\,\,\,4}\end{aligned}} \right)\), and\({\rm{v}} = \left( {\begin{aligned}{{}}1\\1\\1\end{aligned}} \right)\). Verify that 5 is an eigenvalue of \(A\) and \({\rm{v}}\)is an eigenvector. Then orthogonally diagonalize \(A\).
Question: 13. The sample covariance matrix is a generalization of a formula for the variance of a sample of \(N\) scalar measurements, say \({t_1},................,{t_N}\). If \(m\) is the average of \({t_1},................,{t_N}\), then the sample variance is given by
\(\frac{1}{{N - 1}}\sum\limits_{k = 1}^n {{{\left( {{t_k} - m} \right)}^2}} \)
Show how the sample covariance matrix, \(S\), defined prior to Example 3, may be written in a form similar to (1). (Hint: Use partitioned matrix multiplication to write \(S\) as \(\frac{1}{{N - 1}}\) times the sum of \(N\) matrices of size \(p \times p\). For \(1 \le k \le N\), write \({X_k} - M\) in place of \({\hat X_k}\).)
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. \(\left( {\begin{aligned}{{}}{\bf{6}}&{\bf{2}}&{\bf{9}}&{ - {\bf{6}}}\\{\bf{2}}&{\bf{6}}&{ - {\bf{6}}}&{\bf{9}}\\{\bf{9}}&{ - {\bf{6}}}&{\bf{6}}&{\bf{2}}\\{\bf{6}}&{\bf{9}}&{\bf{2}}&{\bf{6}}\end{aligned}} \right)\)
In Exercises 17–24, \(A\) 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 \(m \times n\) “diagonal” matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.
23. Let \(U = \left( {{u_1}...{u_m}} \right)\) and \(V = \left( {{v_1}...{v_n}} \right)\) where the \({{\bf{u}}_i}\) and \({{\bf{v}}_i}\) are in Theorem 10. Show that \(A = {\sigma _1}{u_1}v_1^T + {\sigma _2}{u_2}v_2^T + ... + {\sigma _r}{u_r}v_r^T\).
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