Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Show that similar matrices have the same eigenvalues. Hint: $\vec{v}$ If is an eigenvector of $S^{- 1} AS$ , then role="math" localid="1659529994406" $S \vec{v}$ is an eigenvector of A.

We have proved that similar matrices have the same eigenvalues.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of the Eigenvectors

Eigenvectors are a nonzero vector that is mapped by a given linear transformation of a vector space onto a vector that is the product of a scalar multiplied by the original vector.

Step 2: Find eigenvalue

Assume, that is an Eigen vector for $S^{1} AS$ .

Therefore, by definition:

$S^{- 1} A S^{r} v = λ v^{r}$

Now manipulate the above equation as shown below:

$S^{- 1} AS \hat{v} = {λv}_{r}^{r} {SS}^{- 1} ASv = {Sλ}_{r}^{r} (Multiplyby S fromLeft) ASv = λS \overset{r}{r} (M)$

From the above, the Eigen value of $A$ is $λ$ and $S_{v}^{r}$ is the Eigenvector.

Similarly, Eigen vector of A is $S_{w}^{u}$ then ${S^{- 1}}_{w}^{u}$ is an Eigen vector for $S^{- 1} A S .$

Hence, similar matrices have same eigen values.

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Linear Algebra With Applications

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Short Answer

Show that similar matrices have the same eigenvalues. Hint: $\vec{v}$ If is an eigenvector of $S^{- 1} AS$ , then role="math" localid="1659529994406" $S \vec{v}$ is an eigenvector of A.

Step by Step Solution

Step 1: Definition of the Eigenvectors

Step 2: Find eigenvalue

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Linear Algebra With Applications

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Short Answer

Show that similar matrices have the same eigenvalues. Hint:v→ If is an eigenvector of S-1AS, then role="math" localid="1659529994406" Sv→ is an eigenvector of A.

Step by Step Solution

Step 1: Definition of the Eigenvectors

Step 2: Find eigenvalue

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Show that similar matrices have the same eigenvalues. Hint: $\vec{v}$ If is an eigenvector of $S^{- 1} AS$ , then role="math" localid="1659529994406" $S \vec{v}$ is an eigenvector of A.