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Math 314: Linear Algebra Spring 2006 Name Review: Eigenvalues and eigenvectors 1. Give the definition of eigenvector for a matrix A: 2. Give an example showing why it is useful to know that v is an eigenvector. 3. Given an n × n matrix A what is the characteristic polynomial? What is its degree? 4. Given an n × n matrix A, how many different eigenvalues can you find at most? Justify your answer. 5. Let λ be an eigenvalue for the n×n matrix A. Show that the set of all eigenvectors corresponding to λ together with the zero vector form a subspace for R n . 1 6. Suppose that 2 is a solution of the characteristic polynomial of an n × n matrix A. The reduced echelon for of the matrix A − 2Id must contain a zero row. Why? 7. If zero is an eigenvalue for the n × n matrix then A is not invertible. Why? 8. If λ is an eigenvalue for the n × n matrix A show that λn is an eigenvalue for the matrix An . 9. What are the eigenvalues of a diagonal matrix? 10. Given a n × n diagonal matrix A what is a basis of eigenvectors? 2 11. What does it mean for two n × n matrices A and B to be similar? 12. Show that two similar n × n matrices have the same determinant. 13. Show that the two following matrices are not similar ¶ ¶ µ µ 1 0 1 3 , B= A= 0 1 1 4 1 2 1 A= 0 2 0 0 0 4 1 2 1 A= 0 2 0 0 0 2 3 1 2 1 B = 0 4 0 , 0 0 4 1 2 1 B = 0 1 0 , 0 0 4 14. Let A and B two 4 × 4 matrices. Assume that rank(A) = 1 and rank(B) = 2. Can A and B be similar? 15. Page 307 number 21. 16. Suppose that A is a 6 × 6 matrix with characteristic polynomial cA (λ) = (1 + λ)(1 − λ)2 (2 − λ)3 . (a) Prove that it is impossible to find three linearly independent vectors vi , i = 1, 2, 3, such that Avi = vi , i = 1, 2, 3. (b) If A is diagonalizable, what are the dimensions of the eigenspaces E−1 , E1 , E2 . 17. If A and B are invertible matrices prove that AB and BA are similar. 4