Solving Coupled Tensor Equation $\mathcal{A}⋉\mathcal{X}=\mathcal{B},\mathcal{X}⋉\mathcal{C}=\mathcal{D}$ using Semi-Tensor Products in the t-product

Abstract

This paper investigates the solution of coupled third-order tensor equation $\mathcal{A}⋉\mathcal{X}=\mathcal{B},\mathcal{X}⋉\mathcal{C}=\mathcal{D},$ of arbitrary dimensions by incorporating semi-tensor product (STP) within t-product framework, where the unknown $\mathcal{X}$ can take form of vector, matrix, or tensor. For the unknown $\mathcal{X}$, we establish a necessary and sufficient condition that provides an equivalence criterion for the existence of solutions. Moreover, the explicit structure (Toeplitz, Circulant) of $\mathcal{C}$ and $\mathcal{D}$ is characterized. Theoretical results are supported by several illustrative examples.

31 Pages, 2 figures