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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Question: Find the principal components of the data for Exercise 2.
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 \({\hat X_k}\).)
Compute the quadratic form \({{\bf{x}}^T}A{\bf{x}}\), when \(A = \left( {\begin{aligned}{{}}5&{\frac{1}{3}}\\{\frac{1}{3}}&1\end{aligned}} \right)\) and
a. \({\bf{x}} = \left( {\begin{aligned}{{}}{{x_1}}\\{{x_2}}\end{aligned}} \right)\)
b. \({\bf{x}} = \left( {\begin{aligned}{{}}6\\1\end{aligned}} \right)\)
c. \({\bf{x}} = \left( {\begin{aligned}{{}}1\\3\end{aligned}} \right)\)
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 \(A\).
Let B be an \(n \times n\) symmetric matrix such that \({B^{\bf{2}}} = B\). Any such matrix is called a projection matrix (or an orthogonal projection matrix.) Given any y in \({\mathbb{R}^n}\), let \({\bf{\hat y}} = B{\bf{y}}\)and\({\bf{z}} = {\bf{y}} - {\bf{\hat y}}\).
a) Show that z is orthogonal to \({\bf{\hat y}}\).
b) Let W be the column space of B. Show that y is the sum of a vector in W and a vector in \({W^ \bot }\). Why does this prove that By is the orthogonal projection of y onto the column space of B?
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