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 a dynamical system $\vec{x} (t + 1) = A \vec{x} (t)$ with two components. The accompanying sketch shows the initial state vector ${\vec{x}}_{0}$ and two eigen vectors $\vec{υ_{1}} a n d \vec{υ_{2}}$ of A (with eigen values $\vec{λ_{1}} and \vec{λ_{2}}$ respectively). For the given values of $\vec{λ_{1}} and \vec{λ_{2}}$ , draw a rough trajectory. Consider the future and the past of the system.

$\vec{λ_{1}} = 1, \vec{λ_{2}} = 0.9$

So, the required solution is $A^{t} x_{0} = α υ_{1} + 0 . 9^{t} β υ_{2}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Define the eigenvector

Eigenvector: An eigenvector of $A$ is a nonzero vector $v$ in $R_{n}$ such that $A v = λ v$ , for some scalar $λ$ .

Step 2: Note the given data

It is given that:

$\vec{λ_{1}} = 1, \vec{λ_{2}} = 0.9$

Given graph is:

Step 3: Finding the required solution

We have:

$A υ_{1} = υ_{1} A υ_{2} = 0.9 υ_{2}$

For $x_{0} = α υ_{1} + β υ_{2}$ ,We have:

role="math" localid="1668079897383" $A x_{0} = A (α υ_{1} + β υ_{2}) = α A υ_{1} + β A υ_{2} = α υ_{1} + 0.9 β υ_{2}$

Therefore, $A^{t} x_{0} = α υ_{1} + 0 . 9^{t} β υ_{2}$ .

Hence, the solutions is $A^{t} x_{0} = α υ_{1} + 0 . 9^{t} β υ_{2}$ .

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Linear Algebra and its Applications

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Step 1: Define the eigenvector

Step 2: Note the given data

Step 3: Finding the required solution

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Linear Algebra and its Applications

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Step by Step Solution

Step 1: Define the eigenvector

Step 2: Note the given data

Step 3: Finding the required solution

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