MATH 110

Math 110, Fall 2015. Homework 13, due Nov 25. Prob 1. Let V be a complex n-dimensional space and let T ∈ L(V ) be such that null T n−3 6= null T n−2 . How many distinct eigenvalues can T have? Prob 2. Let V = P3(C) and let D ∈ L(V ) be the differentiation operator. Find a square root of II + D. Prob 3. Let V be a complex (finite-dimensional) vector space and let T ∈ L(V ). Prove that there exist operators D and N in L(V ) such that D is diagonalizable, N is nilpotent, and DN = ND. Prob 4. Suppose that V is a complex vector space of dimension n. Let T ∈ L(V ) be invertible. Let p denote the characteristic polynomial of T and let q denote the characteristic polynomial of T −1 . Prove that q(z) = z n p(0) p 1 z  for all z ∈ C. Prob 5. Suppose the Jordan form of an operator T ∈ L(V ) consists of Jordan blocks of sizes 3 × 3, 4 × 4, 1 × 1, 5 × 5, 2 × 2, corresponding to eigenvalues λ1, λ2, λ3, λ2, λ1, respectively. Assuming that λi 6= λj for i 6= j, find the minimal and the characteristic polynomial of T.