This preview shows page 1. Sign up to view the full content.
Unformatted text preview: be the diagonal entries of the normal form of xI-A . For which matrices A is f 1 6 = 1? 12) Construct T with minimal polynomial x 2 ( x-1) 2 and characteristic polynomial x 3 ( x-1) 4 . Describe the primary decomposition of V under T and nd the projections on the primary components. 12) If N is a nilpotent linear operator on V , show that for any polynomial f the semi-simple part of f ( N ) is a scalar multiple of I . 12) Let F be a subeld of the complex numbers, V a nite-dimensional vectorspace over F and T a semi-simple operator on V . If f is any polynomial over F , prove that f ( T ) is semi-simple. 12) Let T be a linear operator on n-dimensional V over F a subeld of C . Prove that T is semi-simple i if f is a polynomial over F and f ( T ) is nilpotent, then f ( T ) = 0. 1...
View Full Document
This document was uploaded on 01/25/2012.
- Spring '09