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Q7.5-4E

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Linear Algebra and its Applications
Found in: Page 395
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Question: Find the principal components of the data for Exercise 2.

The principal components are \(\left[ {\begin{array}{*{20}{c}}{0.44013}\\{0.897934}\end{array}} \right]{\rm{ and }}\left[ {\begin{array}{*{20}{c}}{ - 0.897934}\\{0.44013}\end{array}} \right]\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: 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 matrix is:

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

Step 2: The Principal Components

From exercise 2, the covariance matrix is:

\(S = \left[ {\begin{array}{*{20}{c}}{5.6}&8\\8&{18}\end{array}} \right]\)

The eigenvalues for this matrix will be:

\(\begin{array}{l}{\lambda _1} = 21.9213\\{\lambda _2} = 1.67874\end{array}\)

The respective eigenvectors are:

\(\begin{array}{l}{x_1} = \left[ {\begin{array}{*{20}{c}}{0.490158}\\1\end{array}} \right]\\{x_2} = \left[ {\begin{array}{*{20}{c}}{ - 2.04016}\\1\end{array}} \right]\end{array}\)

After normalization, we have:

\(\begin{array}{l}{u_1} = \left[ {\begin{array}{*{20}{c}}{0.44013}\\{0.897934}\end{array}} \right]\\{u_2} = \left[ {\begin{array}{*{20}{c}}{ - 0.897934}\\{0.44013}\end{array}} \right]\end{array}\)

Hence, these are the required components.

Most popular questions for Math Textbooks

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).

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

Determine which of the matrices in Exercises 1–6 are symmetric.

1. \(\left[ {\begin{aligned}{{}}3&{\,\,\,5}\\5&{ - 7}\end{aligned}} \right]\)

Construct a spectral decomposition of A from Example 2.

Determine which of the matrices in Exercises 7–12 are orthogonal. If orthogonal, find the inverse.

9. \(\left[ {\begin{aligned}{{}}{ - 4/5}&{\,\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right]\)

Find the matrix of the quadratic form. Assume x is in \({\mathbb{R}^2}\) .

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