/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q38E (M) Orhtogonally diagonalize the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q38E (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.

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\(D = \left( {\begin{aligned}{{}}1&0&0&0\\0&{0.5}&0&0\\0&0&{0.8}&0\\0&0&0&{0.55}\end{aligned}} \right)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Question: 12. Exercises 12鈥�14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Verify the properties of\({A^ + }\):

a. For each\({\rm{y}}\)in\({\mathbb{R}^m}\),\(A{A^ + }{\rm{y}}\)is the orthogonal projection of\({\rm{y}}\)onto\({\rm{Col}}\,A\).

b. For each\({\rm{x}}\)in\({\mathbb{R}^n}\),\({A^ + }A{\rm{x}}\)is the orthogonal projection of\({\rm{x}}\)onto\({\rm{Row}}\,A\).

c. \(A{A^ + }A = A\)and \({A^ + }A{A^ + } = {A^ + }\).

Orthogonally diagonalize the matrices in Exercises 13鈥�22, giving an orthogonal matrix\(P\)and a diagonal matrix\(D\). To save you time, the eigenvalues in Exercises 17鈥�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.

22. \(\left( {\begin{aligned}{{}}4&0&1&0\\0&4&0&1\\1&0&4&0\\0&1&0&4\end{aligned}} \right)\)

Orthogonally diagonalize the matrices in Exercises 13鈥�22, giving an orthogonal matrix\(P\)and a diagonal matrix\(D\). To save you time, the eigenvalues in Exercises 17鈥�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. \(\left( {\begin{aligned}{{}}1&{ - 6}&4\\{ - 6}&2&{ - 2}\\4&{ - 2}&{ - 3}\end{aligned}} \right)\)

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.

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 \(A{A^T}\).)

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

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

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.