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Unformatted text preview: Homework assignment 10 Section 6.3 pp. 197-198 Exercise 1. Let V be a finite dimensional vector space. What is the minimal polynomial for the identity operator on V ? What is the minimal polynomial for the zero operator? Solution: The minimal polynomial for the identity operator is p ( x ) = x- 1 . It is monic, of degree 1 and it annihilates the identity operator. The minimal polynomial for the zero operator is p ( x ) = x . Exercise 3. Let A be the 4 4 real matrix A = 1 1- 1- 1- 2- 2 2 1 1 1- 1 Show that the characteristic polynomial for A is x 2 ( x- 1) 2 and that it is also the minimal polynomial. Solution: Calculating det( xI- A ) we get x 2 ( x- 1) 2 . The minimal polynomial should have the same degree 1 factors, i.e. x and ( x- 1) . Calculating the remain- ing possibilities we get: A ( A- I ) 6 = 0 , A 2 ( A- I ) 6 = 0 , A ( A- 1) 2 6 = 0 . Therefore the minimal polynomial is the characteristic polynomial. Exercise 4. Is the matrix A of Exercise 3 similar over the field of complex numbers to a diagonal matrix? Solution: One can easily check that the matrices A and A- I have rank 3 ....
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This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.
- Spring '10