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Math / Linear Algebra: A Geometric Approach 2 / Chapter 6.1 / Problem 7

Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams

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Suppose x is an eigenvector of A with corresponding eigenvalue . a. Prove that for any

Chapter 6, Problem 7

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QUESTION:

Suppose x is an eigenvector of A with corresponding eigenvalue . a. Prove that for any positive integer n, x is an eigenvector of An with corresponding eigenvalue n. (If you know mathematical induction, this would be a good place to use it.) b. Prove or give a counterexample: x is an eigenvector of A + I . c. If x is an eigenvector of B with corresponding eigenvalue , prove or give a counterexample: x is an eigenvector of A + B with corresponding eigenvalue + . d. Prove or give a counterexample: If is an eigenvalue of A and is an eigenvalue of B, then + is an eigenvalue of A + B.

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QUESTION:

Suppose x is an eigenvector of A with corresponding eigenvalue . a. Prove that for any positive integer n, x is an eigenvector of An with corresponding eigenvalue n. (If you know mathematical induction, this would be a good place to use it.) b. Prove or give a counterexample: x is an eigenvector of A + I . c. If x is an eigenvector of B with corresponding eigenvalue , prove or give a counterexample: x is an eigenvector of A + B with corresponding eigenvalue + . d. Prove or give a counterexample: If is an eigenvalue of A and is an eigenvalue of B, then + is an eigenvalue of A + B.

ANSWER:

Step 1 of 5

(a) When is an eigenvector of with corresponding eigenvalue .

Use Mathematical induction:

By basis of induction:

For ,

Which is true for .

Hence, is true for .

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