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Chapter 7: Q10E (page 395)

Question:Repeat Exercise 9 with \(S = \left( {\begin{array}{*{20}{c}}5&4&2\\4&{11}&4\\2&4&5\end{array}} \right)\).

Short Answer

Expert verified

The variance ofobtained as \({\lambda _1} = 15\).

Step by step solution

01

Mean Deviation form and Covariance Matrix

The Mean Deviation form of any \(p \times N\)is given by:

\(B = \left( {\begin{array}{*{20}{c}}{{{\hat X}_1}}&{{{\hat X}_2}}&{........}&{{{\hat X}_N}}\end{array}} \right)\)

Whose \(p \times p\)covariance matrixis:

\(S = \frac{1}{{N - 1}}B{B^T}\)

02

The Variance

From question, the matrix and the maximum eigenvalue we haveis:

\(\begin{array}{l}S = \left( {\begin{array}{*{20}{c}}5&4&2\\4&{11}&4\\2&4&5\end{array}} \right)\\{\lambda _1} = 15\end{array}\)

The respective unit vector is:

The new variable will be:

Hence, this is the required answer.

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Most popular questions from this chapter

Find the matrix of the quadratic form. Assume x is in \({\mathbb{R}^{\bf{3}}}\).

a. \(3x_1^2 - 2x_2^2 + 5x_3^2 + 4{x_1}{x_2} - 6{x_1}{x_3}\)

b. \(4x_3^2 - 2{x_1}{x_2} + 4{x_2}{x_3}\)

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 \(A = {\sigma _1}{u_1}v_1^T + {\sigma _2}{u_2}v_2^T + ... + {\sigma _r}{u_r}v_r^T\).

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 \({A^ + }A{A^ + } = {A^ + }\).

Question: In Exercises 15 and 16, construct the pseudo-inverse of \(A\). Begin by using a matrix program to produce the SVD of \(A\), or, if that is not available, begin with an orthogonal diagonalization of \({A^T}A\). Use the pseudo-inverse to solve \(A{\rm{x}} = {\rm{b}}\), for \({\rm{b}} = \left( {6, - 1, - 4,6} \right)\) and let \(\mathop {\rm{x}}\limits^\^ \)be the solution. Make a calculation to verify that \(\mathop {\rm{x}}\limits^\^ \) is in Row \(A\). Find a nonzero vector \({\rm{u}}\) in Nul\(A\), and verify that \(\left\| {\mathop {\rm{x}}\limits^\^ } \right\| < \left\| {\mathop {\rm{x}}\limits^\^ + {\rm{u}}} \right\|\), which must be true by Exercise 13(c).

16. \(A = \left( {\begin{array}{*{20}{c}}4&0&{ - 1}&{ - 2}&0\\{ - 5}&0&3&5&0\\{\,\,\,2}&{\,\,0}&{ - 1}&{ - 2}&0\\{\,\,\,6}&{\,\,0}&{ - 3}&{ - 6}&0\end{array}} \right)\)

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix\(P\)and a diagonal matrix\(D\). To save you time, the eigenvalues in Exercises 17–22 are: (17)\( - {\bf{4}}\), 4, 7; (18)\( - {\bf{3}}\),\( - {\bf{6}}\), 9; (19)\( - {\bf{2}}\), 7; (20)\( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

13. \(\left( {\begin{aligned}{{}}3&1\\1&{\,\,3}\end{aligned}} \right)\)
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