This was originally tagged fa.functional analysis, I think. So he's an Operator Algebraic answer. I'm going to make the strong assumption that E is self-adjoint (i.e. closed under taking the hermitian transpose). If not, then really this is an algebraic question, and it's probably irrelevant that you are working with the complex numbers...

Anyway, then E is a finite-dimensional von Neumann algebra. The action of E on M_n is the same as identifying M_n with $\mathbb C^n \otimes \mathbb C^n = \ell^2_n \otimes \ell^2_n$ and letting E act as $E \otimes 1$. Then invariant subspaces for $E$ correspond to orthongonal projections in the commutant of E, which by Tomita is $E' \otimes M_n$ where $E' = \{ A\in M_n : AB=BA (B\in E)\}$ the commutant of $E$ in $M_n$. We identify $E'\otimes M_n$ with $M_n(E')$, and then it's just (ahem!) a case of working out the projections (self-adjoint idempotents) here. In concrete cases, this is probably not too hard...

guessis that this is quite a specialised question (as opposed to looking at E acting on C^n). Phil: do you have a more specific question?? – Matthew Daws Jun 23 '10 at 6:52