t-Product and t-STP of Cubic Matrices With Application to Hyper-Networked Systems

TL;DR

This paper introduces new algebraic operations on cubic matrices, enabling the modeling of coupled dynamics in hyper-networked systems, with applications to control systems and evolutionary game modeling.

Contribution

It develops the t-semi-tensor product (t-STP), unifying previous products to model interactions across subsystems in cubic matrices.

Findings

t-STP allows for coupled subsystem dynamics.

The algebraic structures of t-STP include groups, rings, and Lie groups.

Application to hyper-networked evolutionary games demonstrates practical utility.

Abstract

Motivated by the study of dynamic control systems, this paper proposes novel algebraic operations on cubic matrices to construct both linear and nonlinear controlled dynamics. The standard t-product of cubic matrices imposes strict dimensional constraints; to resolve this, we first introduce the dimension-keeping semi-tensor product (DK-STP), which generalizes the matrix product for arbitrary dimensions. However, the DK-STP yields decoupled subsystem dynamics because it fails to capture interactions across subsystems corresponding to frontal slices. To overcome this limitation, we propose the t-semi-tensor product (t-STP), an integration of the t-product and the DK-STP that allows for coupled subsystems and greater modeling flexibility. We systematically study the algebraic structures derived from the t-STP over cubic matrices, including groups, rings, modules, and Lie groups. Finally, we obtain t-STP-based dynamic control systems over cubic matrices and demonstrate the utility of this framework by applying it to a hyper-networked evolutionary game modeling supply chain interactions.