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Q7E

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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

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

7. \(\left( {\begin{aligned}{{}{}}{.6}&{\,\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\)

\(P\) is an orthogonal matrix and\({P^{ - 1}} = \left( {\begin{aligned}{{}{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Find the characteristic equation

A matrix\(P\) with, \(n \times n\) dimension, is orthogonal if it satisfies the equation\({P^T}P = {I_n}\).

It is given that \(P = \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\). Find the matrix\({P^T}P\)as shown below:

\(\begin{aligned}{}{P^T}P &= \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}1&0\\0&1\end{aligned}} \right)\\ &= {I_2}\end{aligned}\)

Step 2: Find the inverse

As \({P^T}P = {I_2}\), it can be concluded that \(P\) is an orthogonal matrix. So, the inverse of matrix \(P\) is \({P^T}\). Find \({P^T}\), as follows:

\(\begin{aligned}{}{P^{ - 1}} &= {P^T}\\ &= \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\end{aligned}\)

Thus, \(P\) is an orthogonal matrix and \({P^{ - 1}} = \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\).

Most popular questions for Math Textbooks

Question: Let \({\bf{X}}\) denote a vector that varies over the columns of a \(p \times N\) matrix of observations, and let \(P\) be a \(p \times p\) orthogonal matrix. Show that the change of variable \({\bf{X}} = P{\bf{Y}}\) does not change the total variance of the data. (Hint: By Exercise 11, it suffices to show that \(tr\left( {{P^T}SP} \right) = tr\left( S \right)\). Use a property of the trace mentioned in Exercise 25 in Section 5.4.)

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.

19. Show that the columns of \(V\) are eigenvectors of \({A^T}A\), the columns of \(U\) are eigenvectors of \(A{A^T}\), and the diagonal entries of \({\bf{\Sigma }}\) are the singular values of \(A\). (Hint: Use the SVD to compute \({A^T}A\) and \(A{A^T}\).)

Question: Find the principal components of the data for Exercise 1.

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

In Exercises 25 and 26, mark each statement True or False. Justify each answer.

26.

  1. There are symmetric matrices that are not orthogonally diagonizable.
  2. b. If \(B = PD{P^T}\), where \({P^T} = {P^{ - {\bf{1}}}}\) and D is a diagonal matrix, then B is a symmetric matrix.
  3. c. An orthogonal matrix is orthogonally diagonizable.
  4. d. The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.
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