nrev4 - Review Problems for Exam Four THEORY 1 Define...

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Unformatted text preview: Review Problems for Exam Four THEORY 1. Define “eigenvalue” and “eigenvector” and show that λ is an eigenvalue for A if and only if det ( A- λI ) = 0. 2. Define “similar” and show that similar matrices have the same characteristic polynomial. 3. Suppose the n × n matrix A has eigenvalues λ 1 ,...,λ n . Derive expressions for the sum and the product of the eigenvalues in terms of the matrix A . 4. Show that if λ is an eigenvalue of a non-singular matrix A, then 1 /λ is an eigenvalue of A- 1 . 5. Show that A and A T have the same eigenvalues. Do they always have the same eigenvectors? 6. Explain why Y = ve λt is a solution of Y = AY when λ is an eigenvalue of A and v is a corresponding eigenvector. 7. Show that if v 1 ,v 2 ,v 3 are eigenvectors for A associated with distinct eigen- values λ 1 ,λ 2 ,λ 3 , then { v 1 ,v 2 ,v 3 } is linearly independent. 8. Show that the matrix A is diagonalizable if and only if R n has a basis of eigenvectors of A ....
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nrev4 - Review Problems for Exam Four THEORY 1 Define...

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