Characteristic and hyperinvariant subspaces over the field GF(2)

Let f be an endomorphism of a vector space V over a field K. An f-invariant subspace X⊆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.