MA502 - Homework 8.
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
A =
0 1 2
0 0 2
0 0 0 !
.
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.
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