Solving Tensor Equation Systems under the Semi-Tensor Product: A t-Product Approach

Fathi, Javad

doi:10.22072/wala.2026.2079726.1481

Solving Tensor Equation Systems under the Semi-Tensor Product: A t-Product Approach

Document Type : Research Paper

Author

Javad Fathi

Department of Mathematics, University of hormozgan, Bandarabbas, Iran

10.22072/wala.2026.2079726.1481

Abstract

This paper addresses the solvability of the tensor system A⋉X = B, X ⋉C = D using the
semi-tensor product framework. Initially, we examine the associated tensor-vector system
under the semi-tensor product operation, deriving necessary and sufficient conditions for
the existence of solutions and propose explicit solution methods. Furthermore, we introduce suitable tensor partitioning techniques that enable the reduction of the tensor-vector
system to a linear system formulated via the t-product. This approach facilitates a comprehensive study of the solvability of the tensor system when the unknown tensor X is a
vector, matrix or a higher-order tensor under the semi-tensor product. We apply the STP
framework to model and control Boolean gene regulatory networks with pinning control
inputs. Several illustrative examples are provided to demonstrate the applicability and effectiveness of the proposed approaches. Our study includes the tensor equation A⋉X = B
and the matrix system A ⋉ X = B, X ⋉ C = D as special cases.

Keywords

Tensor equation

Semi-tensor product

Toeplitz tensor

t-product