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Q37E

Expert-verified
Found in: Page 395

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

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)

See the step by step solution

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 areE = \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 asD = \left( {\begin{aligned}{{}}{19}&0&0&0\\0&{11}&0&0\\0&0&5&0\\0&0&0&{ - 11}\end{aligned}} \right).