Algebra II 189-251B Instructor: Dr. E. Goren.

Assignment 10

To be submitted by April 6, 12:00

1. Let S and T be commuting linear maps from a vector space V to itself. In the last assignment we proved the following:

Let λ be an eigenvalue of T and let E_{λ} be the corresponding eigenspace. Prove that E_{λ} is S invariant. Conclude that if T is diagonalizable with eigenvalues λ_{1}, …, λ_{r}, and therefore

V = E_{λ_{1}} ⊕ ⋯ ⊕ E_{λ_{r}},

we may decompose S as

S = S_{1} ⊕ ⋯ ⊕ S_{r},

where S_{i} : E_{λ_{i}} ⟶ E_{λ_{i}}.

Assume that both S and T are diagonalizable. Prove now that there exists a basis of V in which both T and S are diagonal.

2. Let A be an n × n matrix over an algebraically closed field such that A^{2} is diagonalizable. Prove that if A is a non-singular matrix then also A is a diagonalizable, and provide an example showing this condition is necessary.

3. Determine the possibilities for the Jordan canonical form of a matrix A with characteristic polynomial Δ_{A}(t) = (t − 1)^{6}(t − 2)^{4}(t − 4)^{5} and minimal polynomial m_{A}(t) = (t − 1)^{3}(t − 2)^{2}(t − 4).

4. Prove that the matrix of complex numbers

A = (\begin{matrix}1 & 1 & 5 & 0 & 0 \\ 7 & 1 & 3 & 0 & 0 \\ 2 & 1 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{matrix})

is diagonalizable. Note: you don’t need to diagonalize it to prove it !!