Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Is $\overset{⇀}{v}$ an eigenvector of $A^{3}$ ? If so, what is the eigenvalue?

Yes, the required eigenvalue is $λ^{3}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of eigenvalue

An eigenvalue of A is a scalar $λ$ such that the equation $Av = λv$ has a nontrivial solution.

Step 2: Find the eigenvalue

Assume that, A is an invertible matrix of order $n \times n$ , and $\overset{⇀}{v}$ is an eigenvector of A corresponding to eigenvalue $λ$ .

Is $\overset{⇀}{v}$ an eigenvector of $A^{3}$ or not, and find the eigenvalue of $A^{3}$ .

If $\overset{⇀}{v}$ is an eigenvector of matrix A then,

$A \overset{⇀}{v} = λ \overset{⇀}{v}$

By the properties of eigenvalues and eigenvectors, if $\overset{⇀}{v}$ is an eigenvector of matrix A , then $\overset{⇀}{v}$ is an eigenvector of matrices $A^{2}, A^{3}, . . . . . . A^{n}$ , as shown below.

$A^{2} \overset{⇀}{v} = λ^{2} \overset{⇀}{v} A^{3} \overset{⇀}{v} = λ^{3} \overset{⇀}{v} A^{n} \overset{⇀}{v} = λ^{n} \overset{⇀}{v}$

Where, n is positive integer.

Now look at $A^{3}$ obtained as,

$A^{3} \overset{⇀}{v} = λ \overset{⇀}{v}$

Thus, $\overset{⇀}{v}$ is an eigenvector of $A^{3}$ corresponding to eigenvalue $λ^{3}$ .

Hence, $\overset{⇀}{v}$ is an eigenvector of $A^{3}$ , and the corresponding eigenvalue is $λ^{3}$ .

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Linear Algebra With Applications

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Short Answer

Is $\overset{⇀}{v}$ an eigenvector of $A^{3}$ ? If so, what is the eigenvalue?

Step by Step Solution

Step 1: Definition of eigenvalue

Step 2: Find the eigenvalue

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Is $\overset{⇀}{v}$ an eigenvector of $A^{3}$ ? If so, what is the eigenvalue?