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

• Published in: Annali di matematica pura ed applicata 2023-02, Vol.202 (1), p.77-93
• Main Authors: Al-Dweik, Ahmad Y., Ghanam, Ryad, Thompson, Gerard, Azad, Hassan
• Format: Article
• Language: English
• Subjects: Differential equations; Eigenvectors; Invariants; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrices (mathematics); Matrix algebra; Subspaces; Symmetry
• ISSN: 0373-3114; 1618-1891
• DOI: 10.1007/s10231-022-01233-7

A problem that is frequently encountered in a variety of mathematical contexts is to find the common invariant subspaces of a single or of a 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 Plücker 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 a 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. Our main motivation lies in Lie symmetry, where the invariant subspaces of the adjoint representations for the Lie symmetry algebra of a differential equation must be known explicitly and comprehensively in order to determine all the ideals of the Lie symmetry algebra.