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

Assignment 9

To be submitted by March 30, 12:00

1. Let S be a linear operator and μ_{1}, …, μ_{r} be distinct eigenvalues of S. Let E_{μ_{1}}, …, E_{μ_{r}} be the corresponding eigenspaces. Let v_{i} ∈ E_{μ_{i}} be a non-zero vector. Prove that the set {v_{1}, …, v_{r}} is linearly independent.

\underline{Hint}: Assume linear dependence. Let v_{k} be the first vector that is linearly dependent on the preceding vectors. Write this equation explicitly and apply S. Try to get a linear dependence of some v_{r} (for r < k) on the preceding vectors and conclude a contradiction.

2. Calculate the characteristic and minimal polynomial of the following matrices with real entries. In each case determine the algebraic and geometric multiplicity of each eigenvalue. Decide which matrix is diagonalizable and for that one, say A, find an invertible matrix M such that M^{−1}AM is diagonal.

(\begin{matrix}4 & −2 & 2 \\ 6 & −3 & 4 \\ 3 & −2 & 3 \end{matrix})  (\begin{matrix}3 & −2 & 2 \\ 4 & −4 & 6 \\ 2 & −3 & 5 \end{matrix})

3. For each matrix N in Exercise 2 (considered as a linear transformation T) find the Primary Decomposition, i.e., the factorization of the minimal polynomial, the kernels of the factors, and for each kernel a matrix representation of T.

4. Let V be a vector space and let W_{1}, …, W_{r} be subspaces of V. Prove that the following are equivalent:

(1) Every vector v can be written uniquely as v = w_{1} + ⋯ + w_{r} with w_{i} ∈ W_{i}.

(2) If B_{i} is a basis for W_{i} then ∪_{i=1}^{r}B_{i} is a basis for V.

(3) Every vector v can be written as v = w_{1} + ⋯ + w_{r} with w_{i} ∈ W_{i} and dim(V) = dim(W_{1}) + ⋯ + dim(W_{r}).

5. Let A be a matrix in block form:

A = (\begin{matrix}A_{1} & 0 & ⋯ & 0 \\ 0 & A_{2} \\ . & . & . \\ 0 & 0 & ⋯ & A_{k} \end{matrix}) .

Prove that

Δ_{A} = Δ_{A_{1}}Δ_{A_{2}} ⋯ Δ_{A_{r}},

and

m_{A} = lcm{m_{A_{1}}, m_{A_{2}}, ⋯, m_{A_{r}}}.

You may use the formula

A^{b} = (\begin{matrix}A_{1}^{b} & 0 & ⋯ & 0 \\ 0 & A_{2}^{b} \\ . & . & . \\ 0 & 0 & ⋯ & A_{k}^{b} \end{matrix})

for every positive integer b.

6. Let S and T be commuting linear maps from a vector space V to itself. 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}}.

Remark: This problem will be continued in the next assignment. The final statement will be that two commuting diagonalizable linear operators may be diagonalized simultaneously. This is used very often in applications.