[SOLVED] MA502 Homework 8

Description

Write down detailed proofs of every statement you make

1. Let A be a n×n matrix with a eigenvalue α ∈ C. Set d_i = dim(Ker(A− αI)ⁱ). Let d₀ = 0 and recall that d_k − d_k−1 is the number of Jordan blocks larger or equal than k.
   • If n = 4, and d₁ = 2, d₂ = 4 find the Jordan canonical form of A.
   • If n = 6 and d₁ = 3, d₂ = 5 and d₃ = 6, find the Jordan canonical form of A.
   • If n = 5 and there is one eigenvalue α = 0 with d₁ = 2,d₂ = 3,d₃ = 4; and one eigenvalue α = 1 with d₁ = 1. Find the Jordan canonical form of A.
2. Find all eigenvectors and the size of the Jordan blocks of

.

3. Prove that for any linear transformation A : V → V , with eigenvalues λ₁,…,λ_n and any polynomial f(t) the linear transformation f(A) will have as eigenvalues f(λ₁),…,f(λ_n).
4. Show that if A is a square matrix with zero determinant, then there exists a polynomial p(t) such that

A · p(A) = 0.

5. Find four 4 × 4 matrices A₁,A₂,A₃,A₄ with minimal polynomial of degree 1,2,3,4 respectively.

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