On solutions of matrix equation AXB=C under semi-tensor product

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], has been extensively applied in Boolean control networks, graph colouring, game theory, cryptographic algorithms and so on. In this article, motivated by the existing work by Yao et al. [J Franklin Inst. 2016;353:1109-1131], we further investigate the solvability of the matrix equation AXB=C with respect to semi-tensor product. The case of matrix-vector equation, in which the required unknown X be a vector, is studied first. Compatible condition for matrix dimensions, necessary and sufficient conditions and concrete solving methods are established. Based on this, the solvability of the matrix equation case, in which the unknown X be a matrix, under semi-tensor product is then studied. For each part, several elementary examples are presented to illustrate the efficiency of the results.