The question is as follows:
Does the Weierstrass Canonical form (AKA Kronecker form) of a (linear) matrix pencil require the underlying field to be algebraically closed?
In this context, a (linear) matrix pencil is a matrix-valued function of the form $$ L(\lambda) := \lambda A + B $$ where $A$ and $B$ are square matrices of size $n$. I am particularly interested in regular matrix pencils, i.e. those for which $\det L(\lambda)$ is not identically $0$.
I found a few references online, including this article by Van Dooren, and I'm trying to understand it well enough to apply it to my particular issue. However, I would appreciate it if somebody has a handy reference or quick argument about the underlying field.