From DK-STP to Non-square General Linear Algebras and General Linear Groups

Abstract

A new matrix product, called dimension-keeping semi-tensor product (DK-STP), is proposed. Under DK-STP, the set of $m\times n$ matrices becomes a semi-group $G({m\times n},\mathbb{F})$, and a ring, denoted by $R(m\times n,\mathbb{F})$. Moreover, the Lie bracket can also be defined, which turns the ring into a Lie algebra, called non-square (or STP) general linear algebra, denoted by $\mathrm{gl}(m\times n, \mathbb{F})$. Then the action of semi-group $G(m\times n,\mathbb{F})$ on dimension-free Euclidian space, denoted by $\mathbb{R}^{\infty}$, is discussed. This action leads to discrete-time and continuous time S-systems. Their trajectories are calculated, and their invariant subspaces are revealed. As byproduct of this study, some important concepts for square matrices, such as eigenvalue, eigenvector, determinant, invertibility, etc., have been extended to non-square matrices. Particularly, it is surprising that the famous Cayley-Hamilton theory can also been extended to non-square matrices. Finally, a Lie group, called the non-square (or STP) general Lie group and denoted by $\mathrm{GL}(m\times n,\mathbb{F})$, is constructed, which has $\mathrm{GL}(m\times n,\mathbb{F})$ as its Lie algebra. Their relations with classical Lie group $\mathrm{GL}(m,\mathbb{F})$ and Lie algebra $\mathrm{gl}(m,\mathbb{F})$ are revealed.