/*! 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} Q4E In Exercises 1鈥�4, the matrix A... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q4E (page 267)

In Exercises 1鈥�4, the matrix A is followed by a sequence \(\left\{ {{{\bf{x}}_k}} \right\}\) produced by the power method. Use these data to estimate the largest eigenvalue of A, and give a corresponding eigenvector.

4.\(A = \left( {\begin{aligned}{ {20}{r}}{4.1}&{ - 6}\\3&{ - 4.4}\end{aligned}} \right)\)

\(\left( {\begin{aligned}{ {20}{l}}1\\1\end{aligned}} \right),{\rm{ }}\left( {\begin{aligned}{ {20}{c}}1\\{.7368}\end{aligned}} \right),{\rm{ }}\left( {\begin{aligned}{ {20}{c}}1\\{.7541}\end{aligned}} \right),{\rm{ }}\left( {\begin{aligned}{ {20}{c}}1\\{.7490}\end{aligned}} \right),{\rm{ }}\left( {\begin{aligned}{ {20}{c}}1\\{.7502}\end{aligned}} \right)\)

Short Answer

Expert verified

The largest eigenvalue of A is \( - .4012\) , and the corresponding eigenvector is \(\left( {\begin{aligned}{ {20}{c}}1\\{.7502}\end{aligned}} \right)\).

Step by step solution

01

Given information

A matrix \(A = \left( {\begin{aligned}{ {20}{l}}{4.1}&{ - 6}\\3&{ - 4.4}\end{aligned}} \right)\). A sequence \(\left\{ {{x_k}} \right\}\).

02

Find the Eigenvalue

Compute the value of\(A{x_k}\)and identify the largest entry as follows:

\(A{x_0} = \left( {\begin{aligned}{ {20}{c}}{4.1}&{ - 6}\\3&{ - 4.4}\end{aligned}} \right)\left( {\begin{aligned}{ {20}{l}}1\\1\end{aligned}} \right) = \left( {\begin{aligned}{ {20}{c}}{ - 1.9}\\{ - 1.4}\end{aligned}} \right)\),\({\mu _0} = - 1.9\)

\(A{x_1} = \left( {\begin{aligned}{ {20}{c}}{4.1}&{ - 6}\\3&{ - 4.4}\end{aligned}} \right)\left( {\begin{aligned}{ {20}{c}}1\\{.7368}\end{aligned}} \right) = \left( {\begin{aligned}{ {20}{c}}{ - .3208}\\{ - .24192}\end{aligned}} \right)\),\({\mu _1} = - .3208\)

\(A{x_2} = \left( {\begin{aligned}{ {20}{c}}{4.1}&{ - 6}\\3&{ - 4.4}\end{aligned}} \right)\left( {\begin{aligned}{ {20}{c}}1\\{.7541}\end{aligned}} \right) = \left( {\begin{aligned}{ {20}{c}}{ - .4246}\\{ - .31804}\end{aligned}} \right)\),\({\mu _2} = - .4296\)

\(A{x_3} = \left( {\begin{aligned}{ {20}{c}}{4.1}&{ - 6}\\3&{ - 4.4}\end{aligned}} \right)\left( {\begin{aligned}{ {20}{c}}1\\{.7490}\end{aligned}} \right) = \left( {\begin{aligned}{ {20}{c}}{ - .394}\\{.2956}\end{aligned}} \right)\),\({\mu _3} = - .394\)

\(A{x_4} = \left( {\begin{aligned}{ {20}{c}}{4.1}&{ - 6}\\3&{ - 4.4}\end{aligned}} \right)\left( {\begin{aligned}{ {20}{c}}1\\{.7502}\end{aligned}} \right) = \left( {\begin{aligned}{ {20}{c}}{ - .4012}\\{ - .30088}\end{aligned}} \right)\),\({\mu _4} = - .4012\)

Hence, the largest absolute entry is 0.4012. So, the eigenvalue is equal to 0.4012. The corresponding eigenvector is \(\left( {\begin{aligned}{ {20}{c}}1\\{.7502}\end{aligned}} \right)\).

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: Show that if \(A\) and \(B\) are similar, then \(\det A = \det B\).

Show that if \({\bf{x}}\) is an eigenvector of the matrix product \(AB\) and \(B{\rm{x}} \ne 0\), then \(B{\rm{x}}\) is an eigenvector of\(BA\).

Question: In Exercises 21 and 22, \(A\) and \(B\) are \(n \times n\) matrices. Mark each statement True or False. Justify each answer.

  1. The determinant of \(A\) is the product of the diagonal entries in \(A\).
  2. An elementary row operation on \(A\) does not change the determinant.
  3. \(\left( {\det A} \right)\left( {\det B} \right) = \det AB\)
  4. If \(\lambda + 5\) is a factor of the characteristic polynomial of \(A\), then 5 is an eigenvalue of \(A\).

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

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

For the matrix A, find real closed formulas for the trajectory x鈫�(t+1)=Ax炉(t) where x鈫�=[01]. Draw a rough sketchA=[7-156-11]
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.