The Solution of Tensor Equation A_cX=B via C-Product^*

The solvability conditions, symmetric solutions, and antisymmetric solutions of matrix equations $AX=B$ are important research topics in matrix theory. As a higher-order generalization of matrices, tensors have made the research on solving tensor equations a hot topic in recent years. This paper focuses on the representation, properties, and computational methods of tensor generalized inverses under the C-product, and systematically explores their applications in solving the tensor equation $\mathcal{A}{*}_c\mathcal{X}=\mathcal{B}$. Firstly, the definition, existence conditions, analytical expressions, and computational algorithms of tensor generalized inverses under the C-product are discussed. By applying tensor generalized inverses under the C-product, the solvability conditions of tensor equation $\mathcal{A}{*}_c\mathcal{X}=\mathcal{B}$$\mathcal{A}{*}_c$ are derived. The minimum norm solution method for consistent equation $AX=B$ and the minimal norm least squares solution inconsistent equation $\mathcal{A}{*}_c\mathcal{X}=\mathcal{B}$ are presented, respectively. Finally, numerical experiments were provided to verify the correctness of the theoretical analysis and algorithm implementation through numerical experiments, demonstrating the effectiveness of solving tensor equations under the C-product.