/*! 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] In Exercises 37鈥�40, use a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q38E (page 267)

[M] In Exercises 37鈥�40, use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.

38. \(\left( {\begin{array}{*{20}{c}}9&{ - 4}&{ - 2}&{ - 4}\\{ - 56}&{32}&{ - 28}&{44}\\{ - 14}&{ - 14}&6&{ - 14}\\{42}&{ - 33}&{21}&{ - 45}\end{array}} \right)\)

Short Answer

Expert verified

The eigenvalues are \(\lambda = \left( {13, - 12, - 12,13} \right)\). The basis for eigenspace of \(\lambda = 13\) is \(N = \left\{ {\left( {\begin{array}{*{20}{c}}{ - 1}\\0\\2\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}1\\{ - 4}\\0\\3\end{array}} \right)} \right\}\). The basis for eigenspace of \(\lambda = - 12\) is \(N = \left\{ {\left( {\begin{array}{*{20}{c}}2\\7\\7\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\{ - 1}\\0\\1\end{array}} \right)} \right\}\).

Step by step solution

01

Command for input the matrix in MATLAB

Write the command given below to input the matrix

\( > > {\rm{ A }} = {\rm{ }}\left( {{\rm{9 }} - 4{\rm{ }} - 2\, - {\rm{4}};{\rm{ }} - {\rm{56 32 }} - {\rm{28 }}\,{\rm{44}};{\rm{ }} - 14{\rm{ }} - {\rm{14 6 }} - {\rm{14}};{\rm{ 42 }} - 33{\rm{ 21 }} - 45} \right)\)

02

Command for finding the eigenvalues of the matrix

\( > > {\rm{ ev}} = {\rm{eig}}\left( {\rm{A}} \right)\)

The output will be\({\rm{ev}} = \left( {13, - 12, - 12,13} \right)\).

Hence, the eigenvalues are \({\rm{ev}} = \left( {13, - 12, - 12,13} \right)\).

03

Command for finding the null basis for each eigenvalue

Write the command given below to find the null basis corresponding to \(\lambda = 13\):

\( > > {\rm{ N}} = {\rm{A}} - {\rm{ev}}\left( 1 \right)*{\rm{eye}}\left( 4 \right)\)

The output is given below:

Hence, the basis for eigenspace of \(\lambda = 13\) is \(N = \left\{ {\left( {\begin{array}{*{20}{c}}{ - 1}\\0\\2\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}1\\{ - 4}\\0\\3\end{array}} \right)} \right\}\).

Write the command given below to find thenull basiscorresponding to \(\lambda = - 12\):

\( > > {\rm{ N}} = {\rm{A}} - {\rm{ev}}\left( 2 \right)*{\rm{eye}}\left( 4 \right)\)

The output is given below:

Hence, the basis for eigenspace of \(\lambda = - 12\) is \(N = \left\{ {\left( {\begin{array}{*{20}{c}}2\\7\\7\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}0\\{ - 1}\\0\\1\end{array}} \right)} \right\}\).

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: Find the characteristic polynomial and the eigenvalues of the matrices in Exercises 1-8.

8. \(\left[ {\begin{array}{*{20}{c}}7&- 2\\2&3\end{array}} \right]\)

Question: Diagonalize the matrices in Exercises \({\bf{7--20}}\), if possible. The eigenvalues for Exercises \({\bf{11--16}}\) are as follows:\(\left( {{\bf{11}}} \right)\lambda {\bf{ = 1,2,3}}\); \(\left( {{\bf{12}}} \right)\lambda {\bf{ = 2,8}}\); \(\left( {{\bf{13}}} \right)\lambda {\bf{ = 5,1}}\); \(\left( {{\bf{14}}} \right)\lambda {\bf{ = 5,4}}\); \(\left( {{\bf{15}}} \right)\lambda {\bf{ = 3,1}}\); \(\left( {{\bf{16}}} \right)\lambda {\bf{ = 2,1}}\). For exercise \({\bf{18}}\), one eigenvalue is \(\lambda {\bf{ = 5}}\) and one eigenvector is \(\left( {{\bf{ - 2,}}\;{\bf{1,}}\;{\bf{2}}} \right)\).

14. \(\left( {\begin{array}{*{20}{c}}4&0&{ - 2}\\2&5&4\\0&0&5\end{array}} \right)\)

In Exercises 9鈥�16, find a basis for the eigenspace corresponding to each listed eigenvalue.

10. \(A = \left( {\begin{array}{*{20}{c}}{10}&{ - 9}\\4&{ - 2}\end{array}} \right)\), \(\lambda = 4\)

Question: In Exercises \({\bf{5}}\) and \({\bf{6}}\), the matrix \(A\) is factored in the form \(PD{P^{ - {\bf{1}}}}\). Use the Diagonalization Theorem to find the eigenvalues of \(A\) and a basis for each eigenspace.

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

Question: Construct a random integer-valued \(4 \times 4\) matrix \(A\), and verify that \(A\) and \({A^T}\) have the same characteristic polynomial (the same eigenvalues with the same multiplicities). Do \(A\) and \({A^T}\) have the same eigenvectors? Make the same analysis of a \(5 \times 5\) matrix. Report the matrices and your conclusions.

See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.