Math 110 Fall 2015 Homework 13 Due Nov 25 Prob 1 Let V Be A

Math 110 Fall 2015homework 13 Due Nov 25prob 1 Let V Be A Complex

Let V be a complex n-dimensional space and let T ∈ L(V) be such that null Tⁿ⁻³ ≠ null Tⁿ⁻². How many distinct eigenvalues can T have?

Let V = P₃(C) and let D ∈ L(V) be the differentiation operator. Find a square root of I + D.

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.

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.

Suppose the Jordan form of an operator T ∈ L(V) consists of Jordan blocks of sizes 3, 4, 1, 5, 2, corresponding to eigenvalues λ₁, λ₂, λ₃, λ₂, λ₁, respectively. Assuming that λ_i ≠ λ_j for i ≠ j, find the minimal and the characteristic polynomial of T.