This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Review Problems for Exam Four THEORY 1. Define eigenvalue and eigenvector and show that is an eigenvalue for A if and only if det ( A I ) = 0. 2. Define similar and show that similar matrices have the same characteristic polynomial. 3. Suppose the n n matrix A has eigenvalues 1 ,..., n . Derive expressions for the sum and the product of the eigenvalues in terms of the matrix A . 4. Show that if is an eigenvalue of a nonsingular matrix A, then 1 / 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 t is a solution of Y = AY when 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 1 , 2 , 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 ....
View
Full
Document
 Spring '09
 RUDYAK

Click to edit the document details