From DK-STP to non-square general linear algebra and general linear group

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 $R(m \times n, \mathbb{F})$. Then the action of semi-group $G(m \times n, \mathbb{F})$ on dimension-free Euclidean 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. Through this action, 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 be extended to non-square matrices. Finally, the Lie bracket can also be defined, which turns the set of $m \times n$ matrices into a Lie algebra, called non-square general linear algebra, denoted by $gl(m \times n, \mathbb{F})$. Moreover, a Lie group, called the non-square general Lie group and denoted by $GL(m \times n, \mathbb{F})$, is constructed, which has $gl(m \times n, \mathbb{F})$ as its Lie algebra. Their relationship with classical Lie group $GL(m, \mathbb{F})$ and Lie algebra $gl(m, \mathbb{F})$ has also been revealed.

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Research supported in part by CNSF under Grants 62073315 and 62350037.

Received 4 July 2024

Published 12 July 2024