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Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams
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Prove or give a counterexample: a. A and AT have the same eigenvalues. b. A and AT have

Chapter 6, Problem 10

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QUESTION:

Prove or give a counterexample: a. A and AT have the same eigenvalues. b. A and AT have the same eigenvectors.

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QUESTION:

Prove or give a counterexample: a. A and AT have the same eigenvalues. b. A and AT have the same eigenvectors.

ANSWER:

Problem 10

Prove or give a counterexample:

   a.  and  have the same eigenvalues

   b.  and  have the same eigenvectors

                                                              Step by Step Solution

Step 1 of 2

Suppose  be an eigenvalue of an arbitrary matrix  .

By definition of eigenvalues,  satisfies corresponding characteristic equations. Therefore,

 

Now, it is known that  . Then,

 

Since  , therefore,

 

Hence,  and  have the same eigenvalues.

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