Solving Coupled Tensor Equation $\mathcal{A} \ltimes \mathcal{X} =\mathcal{B}, \ \mathcal{X}\ltimes \mathcal{C}=\mathcal{D}$ using Semi-Tensor Products in the t-product
This work provides a theoretical framework for solving a specific type of coupled tensor equations, which could be useful for researchers working with tensor algebra in various applications.
This paper addresses the problem of solving coupled third-order tensor equations of the form $\\mathcal{A} \\ltimes \\mathcal{X} = \\mathcal{B}$ and $\\mathcal{X} \\ltimes \\mathcal{C} = \\mathcal{D}$ using semi-tensor products within the t-product framework. It establishes a necessary and sufficient condition for the existence of solutions for the unknown $\\mathcal{X}$ and characterizes the explicit structure of $\\mathcal{C}$ and $\\mathcal{D}$.
This paper investigates the solution of coupled third-order tensor equation $\mathcal{A} \ltimes \mathcal{X} = \mathcal{B},\ \mathcal{X} \ltimes \mathcal{C} = \mathcal{D},$ 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.