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Q37E

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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

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. \(\left( {\begin{aligned}{{}}{\bf{6}}&{\bf{2}}&{\bf{9}}&{ - {\bf{6}}}\\{\bf{2}}&{\bf{6}}&{ - {\bf{6}}}&{\bf{9}}\\{\bf{9}}&{ - {\bf{6}}}&{\bf{6}}&{\bf{2}}\\{\bf{6}}&{\bf{9}}&{\bf{2}}&{\bf{6}}\end{aligned}} \right)\)

\(P = \frac{1}{2}\left( {\begin{aligned}{{}}{ - 1}&1&{ - 1}&1\\1&1&{ - 1}&{ - 1}\\{ - 1}&1&1&{ - 1}\\1&1&1&1\end{aligned}} \right)\), \(D = \left( {\begin{aligned}{{}}{19}&0&0&0\\0&{11}&0&0\\0&0&5&0\\0&0&0&{ - 11}\end{aligned}} \right)\)

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1:Find the eigenvalues of the matrix

Use the following MATLAB code to find the eigenvalues of the given matrix:

\(\begin{aligned}{} > > A = \left( {\begin{aligned}{{}}6&2&9&{ - 6}\end{aligned};\,\begin{aligned}{{}}2&6&{ - 6}&9\end{aligned};\,\begin{aligned}{{}}9&{ - 6}&6&2\end{aligned};\,\begin{aligned}{{}}{ - 6}&9&2&6\end{aligned}} \right);\\ > > \left( {\begin{aligned}{{}}{\rm{E}}&{\rm{V}}\end{aligned}} \right) = {\rm{eigs}}\left( A \right);\end{aligned}\)

So, the eigenvalues are\(E = \left( {\begin{aligned}{{}}{19}\\{11}\\5\\{ - 11}\end{aligned}} \right)\).

Step 2: Find the eigenvectors of the matrix

Use the following MATLAB code to find eigenvectors.

\( > > {v_i} = {\rm{nullbasis}}\left( {A - E\left( i \right)*{\rm{eye}}\left( 4 \right)} \right)\)

Following are the eigenvectors of A.

\({v_1} = \left( {\begin{aligned}{{}}{ - 1}\\1\\{ - 1}\\1\end{aligned}} \right)\), \({v_2} = \left( {\begin{aligned}{{}}1\\1\\1\\1\end{aligned}} \right)\), \({v_3} = \left( {\begin{aligned}{{}}{ - 1}\\{ - 1}\\1\\1\end{aligned}} \right)\), and \({v_4} = \left( {\begin{aligned}{{}}1\\{ - 1}\\{ - 1}\\1\end{aligned}} \right)\)

Step 3: Find the orthogonal projection

The orthogonal projections can be calculated as follows:

\(\begin{aligned}{}{{\bf{u}}_1} &= \frac{1}{{\left\| {{v_1}} \right\|}}{v_1}\\ &= \frac{1}{2}\left( {\begin{aligned}{{}}{ - 1}\\1\\{ - 1}\\1\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{ - \frac{1}{2}}\\{\frac{1}{2}}\\{ - \frac{1}{2}}\\{\frac{1}{2}}\end{aligned}} \right)\end{aligned}\)

And,

\(\begin{aligned}{}{{\bf{u}}_2} &= \frac{1}{{\left\| {{v_2}} \right\|}}{v_2}\\ &= \frac{1}{2}\left( {\begin{aligned}{{}}1\\1\\1\\1\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{\frac{1}{2}}\\{\frac{1}{2}}\\{\frac{1}{2}}\\{\frac{1}{2}}\end{aligned}} \right)\end{aligned}\)

And,

\(\begin{aligned}{}{{\bf{u}}_3} &= \frac{1}{{\left\| {{v_3}} \right\|}}{v_3}\\ &= \frac{1}{2}\left( {\begin{aligned}{{}}{ - 1}\\{ - 1}\\1\\1\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{ - \frac{1}{2}}\\{ - \frac{1}{2}}\\{\frac{1}{2}}\\{\frac{1}{2}}\end{aligned}} \right)\end{aligned}\)

And,

\(\begin{aligned}{}{{\bf{u}}_4} &= \frac{1}{{\left\| {{v_4}} \right\|}}{v_4}\\ &= \frac{1}{2}\left( {\begin{aligned}{{}}1\\{ - 1}\\{ - 1}\\1\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{\frac{1}{2}}\\{ - \frac{1}{2}}\\{ - \frac{1}{2}}\\{\frac{1}{2}}\end{aligned}} \right)\end{aligned}\)

Step 4: Find the matrix P and D

The matrix P can be written using orthogonal projections:

\(\begin{aligned}{}P &= \left( {\begin{aligned}{{}}{ - \frac{1}{2}}&{\frac{1}{2}}&{ - \frac{1}{2}}&{\frac{1}{2}}\\{\frac{1}{2}}&{\frac{1}{2}}&{ - \frac{1}{2}}&{ - \frac{1}{2}}\\{ - \frac{1}{2}}&{\frac{1}{2}}&{\frac{1}{2}}&{ - \frac{1}{2}}\\{\frac{1}{2}}&{\frac{1}{2}}&{\frac{1}{2}}&{\frac{1}{2}}\end{aligned}} \right)\\ &= \frac{1}{2}\left( {\begin{aligned}{{}}{ - 1}&1&{ - 1}&1\\1&1&{ - 1}&{ - 1}\\{ - 1}&1&1&{ - 1}\\1&1&1&1\end{aligned}} \right)\end{aligned}\)

The diagonalized matrix can be written as\(D = \left( {\begin{aligned}{{}}{19}&0&0&0\\0&{11}&0&0\\0&0&5&0\\0&0&0&{ - 11}\end{aligned}} \right)\).

Most popular questions for Math Textbooks

(M) Compute an SVD of each matrix in Exercises 26 and 27. Report the final matrix entries accurate to two decimal places. Use the method of Examples 3 and 4.

26. \(A{\bf{ = }}\left( {\begin{array}{*{20}{c}}{ - {\bf{18}}}&{{\bf{13}}}&{ - {\bf{4}}}&{\bf{4}}\\{\bf{2}}&{{\bf{19}}}&{ - {\bf{4}}}&{{\bf{12}}}\\{ - {\bf{14}}}&{{\bf{11}}}&{ - {\bf{12}}}&{\bf{8}}\\{ - {\bf{2}}}&{{\bf{21}}}&{\bf{4}}&{\bf{8}}\end{array}} \right)\)

Show that if A is an \(n \times n\) symmetric matrix, then \(\left( {A{\bf{x}}} \right) \cdot {\bf{y}} = {\bf{x}} \cdot \left( {A{\bf{y}}} \right)\) for x, y in \({\mathbb{R}^n}\).

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.

16. \(\left( {\begin{aligned}{{}}{\,6}&{ - 2}\\{ - 2}&{\,\,\,9}\end{aligned}} \right)\)

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.

17. \(\left( {\begin{aligned}{{}}1&{ - 6}&4\\{ - 6}&2&{ - 2}\\4&{ - 2}&{ - 3}\end{aligned}} \right)\)

Question: 14. 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 \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Given any \({\rm{b}}\) in \({\mathbb{R}^m}\), adapt Exercise 13 to show that \({A^ + }{\rm{b}}\) is the least-squares solution of minimum length. [Hint: Consider the equation \(A{\rm{x}} = {\rm{b}}\), where \(\mathop {\rm{b}}\limits^\^ \) is the orthogonal projection of \({\rm{b}}\) onto Col \(A\).
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