Short Answer

23: Suppose matrix A is similar to B. What is the relationship between the characteristic polynomials of A and B? What does your answer tell you about the eigenvalues of A and B?

Characteristic polynomials of A and B are the same, therefore their eigenvalues are also same.

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TABLE OF CONTENTS

Step 1: Eigenvalues

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled.

Step 2: Relationship between the characteristic polynomials of A and B

If A and B are similar matrices then, $T^{- 1} - A T = B$ , for an invertible T. If det $(A - λ l) = 0$ , then

$0 = d e t^{- 1} d e t (A - λ l) d e t T = d e t (T^{- 1} (A - λ l) T) = d e t (T^{- 1} A T - λ T^{- 1} l T) = d e t (T^{- 1} A T - λ l) = d e t (B - λ l)$

This means that the characteristics polynomials of A and B are the same. From this also follows that their eigenvalues are exactly the same.

Hence, characteristic polynomials of A and B are the same, therefore their eigenvalues are also same.

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Short Answer

23: Suppose matrix A is similar to B. What is the relationship between the characteristic polynomials of A and B? What does your answer tell you about the eigenvalues of A and B?

Step by Step Solution

Step 1: Eigenvalues

Step 2: Relationship between the characteristic polynomials of A and B

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Linear Algebra With Applications

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Short Answer

23: Suppose matrix A is similar to B. What is the relationship between the characteristic polynomials of A and B? What does your answer tell you about the eigenvalues of A and B?

Step by Step Solution

Step 1: Eigenvalues

Step 2: Relationship between the characteristic polynomials of A and B

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