TITLE: Algebra - Decomposition of a matrix polynomial
QUESTION [5 upvotes]: Dear All,
This is related with a problem that I'm trying to solve on my PhD dissertation in econometrics, and I thought that some mathmatician can know the answer. 
What is known about a possible extension, $E$ , of the ring, $ A$ , of all n-by-n matrices with entries in $\mathbb{C}$ such that any non-constant polynomial of $ A[x] $ splits in a product of linear factors in  $E[x]$?
$ax = xa$ iff $a$ is in the commutator of $E$. Moreover, $ A$ is unitary. 
Thanks a lot,
Federico Carlini

REPLY [2 votes]: This is true because of following: any matrix over principal ideal domain can be reduced by elementary Gauss transformations to the Smith normal form (see this: http://en.wikipedia.org/wiki/Smith_normal_form).
The ring of polynomials over any field is such a domain.
Smith normal form can be splitted to linear factors.
Elementary Gauss transformations may be of the form $(z, w)\mapsto (z+p(x)w, w)$, where $p(x)$ is not linear. I think that these transformations cannot be splitted in linear factors (I don't know how to prove it - please let me know if you do know this, I'm curious about).