Write down detailed proofs of every statement you make
1. Let A be a n×n matrix with a eigenvalue α ∈ C. Set di = dim(Ker(A− αI)i). Let d0 = 0 and recall that dk − dk−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,...,λn and any polynomial f(t) the linear transformation f(A) will have as eigenvalues f(λ1),...,f(λn).
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.
