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 Tn−3 6= null Tn−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) = zn

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 .