Characteristic and hyperinvariant subspaces over the field GF(2)

TL;DR

This paper investigates the properties of characteristic and hyperinvariant subspaces over GF(2), providing a new proof of Shoda's theorem that characterizes when characteristic subspaces are not hyperinvariant.

Contribution

It offers a novel proof of a classical theorem, clarifying conditions for the existence of characteristic subspaces that are not hyperinvariant over GF(2).

Findings

Characteristic subspaces can be non-hyperinvariant over GF(2).

A necessary and sufficient condition for such subspaces is established.

The proof simplifies understanding of invariant subspace structures over GF(2).

Abstract

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.