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Q7.6-26 E

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Linear Algebra and its Applications
Found in: Page 267
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Consider an invertible n × n matrix A such that the zero state is a stable equilibrium of the dynamical system x→(t+1)=ATx→(t) What can you say about the stability of the systems.

x→(t+1)=ATx→(t)

The given value is stable

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Define eigen value

Eigenvalues can be used to determine whether a fixed point (also known as an equilibrium point) is stable or unstable.

Finding the stability of the given equation

If the system that involves AT is stable, then all of its eigenvalues are inside the unit circle. since and AT have the exact same eigenvalues then this also applies for AT meaning that the system that involves AT is stable.

Most popular questions for Math Textbooks

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 \(\left( {{\bf{ - 2,}}\;{\bf{1,}}\;{\bf{2}}} \right)\).

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

Define \(T:{{\rm P}_2} \to {\mathbb{R}^3}\) by \(T\left( {\bf{p}} \right) = \left( {\begin{aligned}{{\bf{p}}\left( { - 1} \right)}\\{{\bf{p}}\left( 0 \right)}\\{{\bf{p}}\left( 1 \right)}\end{aligned}} \right)\).

  1. Find the image under\(T\)of\({\bf{p}}\left( t \right) = 5 + 3t\).
  2. Show that \(T\) is a linear transformation.
  3. Find the matrix for \(T\) relative to the basis \(\left\{ {1,t,{t^2}} \right\}\)for \({{\rm P}_2}\)and the standard basis for \({\mathbb{R}^3}\).

In Exercises 9–16, 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: Is \(\lambda = 2\) an eigenvalue of \(\left( {\begin{array}{*{20}{c}}3&2\\3&8\end{array}} \right)\)? Why or why not?

Let \(A = \left( {\begin{aligned}{*{20}{c}}{.4}&{ - .3}\\{.4}&{1.2}\end{aligned}} \right)\). Explain why \({A^k}\) approaches \(\left( {\begin{aligned}{*{20}{c}}{ - .5}&{ - .75}\\1&{1.5}\end{aligned}} \right)\) as \(k \to \infty \).
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