Basis for the linear space of matrices under equivalence

TL;DR

This paper develops a concise basis for the quotient space of matrices under a specific equivalence relation, revealing that its associated Lie algebra is countably infinite dimensional, building on the semi-tensor product framework.

Contribution

It introduces a new concise basis for the matrix quotient space under an equivalence relation, expanding the algebraic understanding of the structure.

Findings

The quotient space has a countably infinite dimensional Lie algebra.

A new basis simplifies the structure of the quotient space.

The work extends the semi-tensor product framework to new algebraic insights.

Abstract

The semi-tensor product (STP) of matrices which was proposed by Daizhan Cheng in 2001 [2], is a natural generalization of the standard matrix product and well defined at every two finite-dimensional matrices. In 2016, Cheng proposed a new concept of semi-tensor addition (STA) which is a natural generalization of the standard matrix addition and well defined at every two finite-dimensional matrices with the same ratio between the numbers of rows and columns [1]. In addition, an identify equivalence relation between matrices was defined in [1], STP and STA were proved valid for the corresponding identify equivalence classes, and the corresponding quotient space was endowed with an algebraic structure and a manifold structure. In this follow-up paper, we give a new concise basis for the quotient space, which also shows that the Lie algebra corresponding to the quotient space is of countably infinite dimension.