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(M) 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.

38. \(\left( {\begin{aligned}{{}}{.{\bf{63}}}&{ - .{\bf{18}}}&{ - .{\bf{06}}}&{ - .{\bf{04}}}\\{ - .{\bf{18}}}&{.{\bf{84}}}&{ - .{\bf{04}}}&{.{\bf{12}}}\\{ - .{\bf{06}}}&{ - .{\bf{04}}}&{.{\bf{72}}}&{ - .{\bf{12}}}\\{ - .{\bf{04}}}&{.{\bf{12}}}&{ - .{\bf{12}}}&{.{\bf{66}}}\end{aligned}} \right)\)

Short Answer

Expert verified

\(P = \frac{1}{5}\left( {\begin{aligned}{{}}{ - 2}&4&2&{ - 1}\\4&2&{ - 1}&{ - 2}\\{ - 1}&2&{ - 4}&2\\2&1&2&4\end{aligned}} \right)\), \(D = \left( {\begin{aligned}{{}}1&0&0&0\\0&{0.5}&0&0\\0&0&{0.8}&0\\0&0&0&{0.55}\end{aligned}} \right)\)

Step by step solution

01

Step 1:Find the eigen values of the matrix

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

\(\begin{aligned}{} > > A = \left( \begin{aligned}{}\begin{aligned}{{}}{.63}&{ - .18}&{ - .06}&{ - .04}\end{aligned};\,\begin{aligned}{{}}{\, - .18}&{.84}&{ - .04}&{.12}\end{aligned};\,\begin{aligned}{{}}{\, - .06}&{ - .04}&{.72}&{ - .12}\end{aligned};\\\,\begin{aligned}{{}}{ - .04}&{.12}&{.12}&{.66}\end{aligned}\end{aligned} \right);\\ > > \left( {\begin{aligned}{{}}{\rm{E}}&{\rm{V}}\end{aligned}} \right) = {\rm{eigs}}\left( A \right);\end{aligned}\)

So, the eigenvalues are\(E = \left( {\begin{aligned}{{}}1\\{0.5}\\{0.8}\\{0.55}\end{aligned}} \right)\).

02

Find the eigen vectors 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.

\({v_1} = \left( {\begin{aligned}{{}}{ - 1}\\2\\{ - \frac{1}{2}}\\1\end{aligned}} \right)\), \({v_2} = \left( {\begin{aligned}{{}}4\\2\\2\\1\end{aligned}} \right)\), \({v_3} = \left( {\begin{aligned}{{}}{ - 1}\\{ - \frac{1}{2}}\\{ - 2}\\1\end{aligned}} \right)\), and \({v_4} = \left( {\begin{aligned}{{}}{ - \frac{1}{4}}\\{ - \frac{1}{2}}\\{\frac{1}{2}}\\1\end{aligned}} \right)\)

03

Find the orthogonal projection

The orthogonal projections can be calculated as follows:

\(\begin{aligned}{}{{\bf{u}}_1} &= \frac{1}{{\left\| {{v_1}} \right\|}}{v_1}\\ &= \frac{1}{5}\left( {\begin{aligned}{{}}{ - 2}\\4\\{ - 1}\\2\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{ - \frac{2}{5}}\\{\frac{4}{5}}\\{ - \frac{1}{5}}\\{\frac{2}{5}}\end{aligned}} \right)\end{aligned}\)

And,

\(\begin{aligned}{}{{\bf{u}}_2} &= \frac{1}{{\left\| {{v_2}} \right\|}}{v_2}\\ &= \frac{1}{5}\left( {\begin{aligned}{{}}4\\2\\2\\1\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{\frac{4}{5}}\\{\frac{2}{5}}\\{\frac{2}{5}}\\{\frac{1}{5}}\end{aligned}} \right)\end{aligned}\)

And,

\(\begin{aligned}{}{{\bf{u}}_3} &= \frac{1}{{\left\| {{v_3}} \right\|}}{v_3}\\ &= \frac{1}{5}\left( {\begin{aligned}{{}}2\\{ - 1}\\{ - 4}\\2\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{\frac{2}{5}}\\{ - \frac{1}{5}}\\{ - \frac{4}{5}}\\{\frac{2}{5}}\end{aligned}} \right)\end{aligned}\)

And,

\(\begin{aligned}{}{{\bf{u}}_4} &= \frac{1}{{\left\| {{v_4}} \right\|}}{v_4}\\ &= \frac{1}{5}\left( {\begin{aligned}{{}}{ - 1}\\{ - 2}\\2\\4\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{ - \frac{1}{5}}\\{ - \frac{2}{5}}\\{\frac{2}{5}}\\{\frac{4}{5}}\end{aligned}} \right)\end{aligned}\)

04

Find the matrix P and D

The matrix P can be written using orthogonal projections as:

\(\begin{aligned}{}P &= \left( {\begin{aligned}{{}}{ - \frac{2}{5}}&{\frac{4}{5}}&{\frac{2}{5}}&{ - \frac{1}{5}}\\{\frac{4}{5}}&{\frac{2}{5}}&{ - \frac{1}{5}}&{ - \frac{2}{5}}\\{ - \frac{1}{5}}&{\frac{2}{5}}&{ - \frac{4}{5}}&{\frac{2}{5}}\\{\frac{2}{5}}&{\frac{1}{5}}&{\frac{2}{5}}&{\frac{4}{5}}\end{aligned}} \right)\\ &= \frac{1}{5}\left( {\begin{aligned}{{}}{ - 2}&4&2&{ - 1}\\4&2&{ - 1}&{ - 2}\\{ - 1}&2&{ - 4}&2\\2&1&2&4\end{aligned}} \right)\end{aligned}\)

The diagonalized matrix can be written as\(D = \left( {\begin{aligned}{{}}1&0&0&0\\0&{0.5}&0&0\\0&0&{0.8}&0\\0&0&0&{0.55}\end{aligned}} \right)\).

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Most popular questions from this chapter

In Exercises 17โ€“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 \(m \times n\) โ€œdiagonalโ€ matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

22. Show that if \(A\) is an \(n \times n\) positive definite matrix, then an orthogonal diagonalization \(A = PD{P^T}\) is a singular value decomposition of \(A\).

In Exercises 17โ€“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 \(m \times n\) โ€œdiagonalโ€ 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 \(A = {\sigma _1}{u_1}v_1^T + {\sigma _2}{u_2}v_2^T + ... + {\sigma _r}{u_r}v_r^T\).

Let B be an \(n \times n\) symmetric matrix such that \({B^{\bf{2}}} = B\). Any such matrix is called a projection matrix (or an orthogonal projection matrix.) Given any y in \({\mathbb{R}^n}\), let \({\bf{\hat y}} = B{\bf{y}}\)and\({\bf{z}} = {\bf{y}} - {\bf{\hat y}}\).

a) Show that z is orthogonal to \({\bf{\hat y}}\).

b) Let W be the column space of B. Show that y is the sum of a vector in W and a vector in \({W^ \bot }\). Why does this prove that By is the orthogonal projection of y onto the column space of B?

Question 8: Use Exercise 7 to show that if A is positive definite, then A has a LU factorization, \(A = LU\), where U has positive pivots on its diagonal. (The converse is true, too).

(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.

27. \(A{\bf{ = }}\left( {\begin{array}{*{20}{c}}{\bf{6}}&{ - {\bf{8}}}&{ - {\bf{4}}}&{\bf{5}}&{ - {\bf{4}}}\\{\bf{2}}&{\bf{7}}&{ - {\bf{5}}}&{ - {\bf{6}}}&{\bf{4}}\\{\bf{0}}&{ - {\bf{1}}}&{ - {\bf{8}}}&{\bf{2}}&{\bf{2}}\\{ - {\bf{1}}}&{ - {\bf{2}}}&{\bf{4}}&{\bf{4}}&{ - {\bf{8}}}\end{array}} \right)\)

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