Dimension-Free Matrix Spaces

TL;DR

This paper introduces dimension-free matrix spaces derived from semi-tensor products, extending Lie algebra structures to infinite-dimensional spaces with properties similar to finite-dimensional Lie algebras.

Contribution

It proposes a novel quotient space framework called dimension-free matrix spaces and extends Lie algebra structures to these infinite-dimensional spaces.

Findings

Dimension-free matrix spaces are constructed as quotient spaces under matrix equivalences.

The Lie bracket structure of general linear algebra is extended to these spaces.

DFGLAs exhibit properties akin to finite-dimensional Lie algebras despite being infinite-dimensional.

Abstract

Based on various types of semi-tensor products of matrices, the corresponding equivalences of matrices are proposed. Then the corresponding vector space structures are obtained as the quotient spaces under equivalences, which are called the dimension-free Matrix spaces (DFESs). Certain structures and properties are investigated. Finaly, the Lie bracket structure of general linear algebra is extended to DfMSs to make them Lie algebras, called dimension-free general linear algebra (DFGLA). Inspire of the fact that the DFGLAs are of infinite dimension, they have most properties of finite dimensional Lie algebras, whicl are studied in the paper.