  Let $f$ be an endomorphism of a vector space $V$ over a field $K$. An
$f$-invariant subspace $X \subseteq V$ is called hyperinvariant (respectively
characteristic) if $X$ is invariant under all endomorphisms (respectively
automorphisms) that commute with $f$. If $|K| > 2$ then all characteristic
subspaces are hyperinvariant. If $|K| = 2$ then there are endomorphisms $f$
with invariant subspaces that are characteristic but not hyperinvariant. In
this paper we give a new proof of a theorem of Shoda, which provides a
necessary and sufficient condition for the existence of characteristic
non-hyperinvariant subspaces.
