2023/05/31 by Daizhan Cheng, Cheng, Daizhan · 2 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Protein Structure and Dynamics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2305.19794
openalex publication_date 2023/05/31 · openalex created_date 2023/06/02 · openalex updated_date 2026/07/28
A new matrix product, called dimension-keeping semi-tensor product (DK-STP), is proposed. Under DK-STP, the set of m× n matrices becomes a semi-group G(m× n,\mathbbF), and a ring, denoted by R(m× n,\mathbbF). 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 gl(m× n, \mathbbF). Then the action of semi-group G(m× n,\mathbbF) on dimension-free Euclidian space, denoted by ℝ∞, 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 GL(m× n,\mathbbF), is constructed, which has GL(m× n,\mathbbF) as its Lie algebra. Their relations with classical Lie group GL(m,\mathbbF) and Lie algebra gl(m,\mathbbF) are revealed.