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Chapter 7: Q39E (page 395)

(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.

39. \(\left( {\begin{aligned}{{}}{.{\bf{31}}}&{.{\bf{58}}}&{.{\bf{08}}}&{.{\bf{44}}}\\{.{\bf{58}}}&{ - .{\bf{56}}}&{.{\bf{44}}}&{ - .{\bf{58}}}\\{.{\bf{08}}}&{.{\bf{44}}}&{.{\bf{19}}}&{ - .{\bf{08}}}\\{ - .{\bf{44}}}&{ - .{\bf{58}}}&{ - .{\bf{08}}}&{.{\bf{31}}}\end{aligned}} \right)\)

Short Answer

Expert verified

\(P = \left( {\begin{aligned}{{}}{\frac{1}{{\sqrt 2 }}}&{\frac{3}{{\sqrt {50} }}}&{ - \frac{2}{5}}&{ - \frac{2}{5}}\\0&{\frac{4}{{\sqrt {50} }}}&{ - \frac{1}{5}}&{\frac{4}{5}}\\0&{\frac{4}{{\sqrt {50} }}}&{\frac{4}{5}}&{ - \frac{1}{5}}\\{\frac{1}{{\sqrt 2 }}}&{ - \frac{3}{{\sqrt {50} }}}&{\frac{2}{5}}&{\frac{2}{5}}\end{aligned}} \right)\), \(D = \left( {\begin{aligned}{{}}{ - 1.25}&0&0&0\\0&{0.75}&0&0\\0&0&{0.75}&0\\0&0&0&0\end{aligned}} \right)\)

Step by step solution

01

Step 1:Findthe eigenvalues of the matrix

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

\(\begin{aligned}{} > > A = \left( \begin{aligned}{}\begin{aligned}{{}}{.31}&{.58}&{.08}&{.44}\end{aligned};\,\begin{aligned}{{}}{\,.58}&{ - .56}&{.44}&{ - .58}\end{aligned};\,\begin{aligned}{{}}{\,.08}&{.44}&{.19}&{ - .08}\end{aligned};\\\,\begin{aligned}{{}}{ - .44}&{.58}&{ - .08}&{.31}\end{aligned}\end{aligned} \right);\\ > > \left( {\begin{aligned}{{}}{\rm{E}}&{\rm{V}}\end{aligned}} \right) = {\rm{eigs}}\left( A \right);\end{aligned}\)

So, the eigenvalues are\(E = \left( {\begin{aligned}{{}}{ - 1.25}\\{0.75}\\{0.75}\\0\end{aligned}} \right)\).

02

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.

\({v_1} = \left( {\begin{aligned}{{}}1\\0\\0\\1\end{aligned}} \right)\), \({v_2} = \left( {\begin{aligned}{{}}3\\2\\2\\0\end{aligned}} \right)\), \({v_3} = \left( {\begin{aligned}{{}}1\\0\\0\\1\end{aligned}} \right)\), and \({v_4} = \left( {\begin{aligned}{{}}3\\4\\4\\{ - 3}\end{aligned}} \right)\)

03

Step 3:Find the orthogonal projection

The orthogonal projections can be calculated as follows:

\(\begin{aligned}{}{{\bf{u}}_1} &= \frac{1}{{\left\| {{v_1}} \right\|}}{v_1}\\ &= \left( {\begin{aligned}{{}}{\frac{1}{{\sqrt 2 }}}\\0\\0\\{\frac{1}{{\sqrt 2 }}}\end{aligned}} \right)\end{aligned}\)

And,

\(\begin{aligned}{}{{\bf{u}}_2} &= \frac{1}{{\left\| {{v_2}} \right\|}}{v_2}\\ &= \left( {\begin{aligned}{{}}{\frac{3}{{\sqrt {50} }}}\\{\frac{4}{{\sqrt {50} }}}\\{\frac{4}{{\sqrt {50} }}}\\{ - \frac{3}{{\sqrt {50} }}}\end{aligned}} \right)\end{aligned}\)

And,

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

And,

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

04

Step 4:Find the matrix P and D 

The matrix P can be written using orthogonal projections as:

\(P = \left( {\begin{aligned}{{}}{\frac{1}{{\sqrt 2 }}}&{\frac{3}{{\sqrt {50} }}}&{ - \frac{2}{5}}&{ - \frac{2}{5}}\\0&{\frac{4}{{\sqrt {50} }}}&{ - \frac{1}{5}}&{\frac{4}{5}}\\0&{\frac{4}{{\sqrt {50} }}}&{\frac{4}{5}}&{ - \frac{1}{5}}\\{\frac{1}{{\sqrt 2 }}}&{ - \frac{3}{{\sqrt {50} }}}&{\frac{2}{5}}&{\frac{2}{5}}\end{aligned}} \right)\)

The diagonalized matrix can be written as\(D = \left( {\begin{aligned}{{}}{ - 1.25}&0&0&0\\0&{0.75}&0&0\\0&0&{0.75}&0\\0&0&0&0\end{aligned}} \right)\).

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Most popular questions from this chapter

Question: 4. Let A be an \(n \times n\) symmetric matrix.

a. Show that \({({\rm{Col}}A)^ \bot } = {\rm{Nul}}A\). (Hint: See Section 6.1.)

b. Show that each y in \({\mathbb{R}^n}\) can be written in the form \(y = \hat y + z\), with \(\hat y\) in \({\rm{Col}}A\) and z in \({\rm{Nul}}A\).

In Exercises 17鈥�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 \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

20. Show that if\(A\)is an orthogonal\(m \times m\)matrix, then \(PA\) has the same singular values as \(A\).

Make a change of variable, \({\bf{x}} = P{\bf{y}}\), that transforms the quadratic form \(x_{\bf{1}}^{\bf{2}} + {\bf{10}}{x_{\bf{1}}}{x_{\bf{2}}} + x_{\bf{2}}^{\bf{2}}\) into a quadratic form with no cross-product term. Give P and the new quadratic form.

Question: 11. Given multivariate data \({X_1},................,{X_N}\) (in \({\mathbb{R}^p}\)) in mean deviation form, let \(P\) be a \(p \times p\) matrix, and define \({Y_k} = {P^T}{X_k}{\rm{ for }}k = 1,......,N\).

  1. Show that \({Y_1},................,{Y_N}\) are in mean-deviation form. (Hint: Let \(w\) be the vector in \({\mathbb{R}^N}\) with a 1 in each entry. Then \(\left( {{X_1},................,{X_N}} \right)w = 0\) (the zero vector in \({\mathbb{R}^p}\)).)
  2. Show that if the covariance matrix of \({X_1},................,{X_N}\) is \(S\), then the covariance matrix of \({Y_1},................,{Y_N}\) is \({P^T}SP\).

Question: 3. Let A be an \(n \times n\) symmetric matrix of rank r. Explain why the spectral decomposition of A represents A as the sum of r rank 1 matrices.
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