Self-inverse linear subspaces of matrices

Abstract

We study linear subspaces of matrices whose inverse spaces are also linear. Based on the fact that any linear space containing the identity matrix and whose inverse space is linear must be a self-inverse space, we introduce such spaces as self-inverse spaces. In fact, as we will show, self-inverse spaces are finite-dimensional complex unitary Jordan algebras. We provide an algebraic classification of all self-inverse spaces of small type and a classification of small-dimensional self-inverse spaces up to isomorphism.