A novel procedure for constructing invariant subspaces of a set of matrices

TL;DR

This paper introduces a new method for constructing common invariant subspaces of matrices using eigenvectors of exterior powers and Plücker relations, with computational implementation in Maple.

Contribution

It presents a novel procedure that definitively finds invariant subspaces by leveraging eigenvectors of exterior powers and algebraic relations, advancing the computational approach.

Findings

Method successfully finds invariant subspaces in complex examples.

Eigenvector computation via exterior powers is effective.

Implementation in Maple automates tedious calculations.

Abstract

A problem that is frequently encountered in a variety of mathematical contexts, is to find the common invariant subspaces of a single, or set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea consists of finding common eigenvectors for exterior powers of the matrices concerned. A convenient formulation of the Pl\"ucker relations is then used to ensure that these eigenvectors actually correspond to subspaces or provide the initial constraints for eigenvectors involving parameters. A procedure for computing the divisors of totally decomposable vector is also provided. Several examples are given for which the calculations are too tedious to do by hand and are performed by coding the conditions found into Maple.