Solvability of matrix equations , under semi-tensor product

In this paper we consider the solvability of the matrix equations , with respect to the semi-tensor product. Using the concept of the semi-tensor product which was initially proposed by Cheng et al. [An introduction to Semi-Tensor product of matrices and its applications, World Scientific, 2012], firstly we investigate the matrix–vector equations , under the semi-tensor product, establish some compatible conditions for the matrices, and propose a necessary and sufficient condition for the solvability of the matrix–vector equation. We then explore the solvability of the matrix equations , under the semi-tensor product. Some elementary examples are given to show the efficiency of the results.