Enumerating Invariant Subspaces of ${\mathbb R}^n$

TL;DR

This paper presents an algorithm to determine all possible counts of invariant subspaces for linear operators on real vector spaces, with potential extensions to other fields.

Contribution

It introduces a novel algorithm for enumerating the number of invariant subspaces of linear operators on real vector spaces.

Findings

Algorithm successfully computes all possible numbers of invariant subspaces.

Provides insights into the structure of invariant subspaces for linear operators.

Discusses potential extensions to vector spaces over arbitrary fields.

Abstract

In this article, we develop an algorithm to calculate the set of all integers $m$ for which there exists a linear operator $T$ on ${\mathbb R}^n$ such that ${\mathbb R}^n$ has exactly $m$ $T$-invariant subspaces. A brief discussion is included as how these methods might be extended to vector spaces over arbitrary fields.