189-251B: Algebra 2 Assignment 7

Due: Wednesday, February 26

1. Let T : V ⟶ V be a linear transformation of finite-dimensional vector spaces over a field F which is not necessarily algebraically closed, and let p(x) = p_{1}(x) ⋯ p_{r}(x) be the factorisation of p(x) into monic irreducible factors.

a) Assume that the factors p_{j}(x) are all distinct. Show that there is a direct sum decomposition

V = V_{1} ⊕ ⋯ ⊕ V_{r}

into F -vector subspaces V_{j} which are stable under T, and such that the restriction T_{j} of T to V_{j} has p_{j}(x) as its minimal polynomial.

b) If dim_{F} (V_{j}) = deg(p_{j}), show that V_{j} cannot be further decomposed into a direct sum of T -stable F -vector subspaces.

2. Let f(x) = a_{0} + a_{1}x + a_{2}x^{2} + ⋯ + a_{k}x^{k} + ⋯ be a power series in x with complex coefficients and suppose that λ ∈ C belongs to the disk of absolute convergence for f. Let M(λ, d) be the d × d Jordan matrix with eigenvalue λ:

M(λ, d) = ( \begin{matrix}λ & 1 & 0 & 0 & ⋯ & 0 \\ 0 & λ & 1 & 0 & ⋯ & 0 \\ 0 & 0 & λ & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & 0 & ⋯ & λ \end{matrix} ) .

Show that the infinite sum

f(M(λ, d)) = a_{0} + a_{1}M(λ, d) + a_{2}M(λ, d)^{2} + ⋯

converges to a d×d matrix, and give a closed form expression for this matrix. (You may find it helpful to consider first the simplest cases where d = 2 and d = 3 in order to guess the general pattern.)

3. Let e^{x} denote the exponential function defined by the power series

e^{x} = 1 + x + \frac{x^{2}}{2!} + ⋯ + \frac{x^{k}}{k!} + ⋯

Using exercise 2 and the theory of the Jordan canonical form, show that the infinite sum defining e^{M} converges, for any d × d matrix M with complex entries, and describe an algorithm to compute it. Use your algorithm to

compute the exponential of the 2 × 2 matrix ( \begin{matrix}1 & −1 \\ 1 & 3 \end{matrix} ).