On solutions of the matrix equation AX=B with respect to semi-tensor product

This paper studies the solutions of the matrix equation AX=B with respect to semi-tensor product. Firstly, the matrix-vector equation AX=B with semi-tensor product is discussed. Compatible conditions are established for the matrices, and a necessary and sufficient condition for the solvability of the matrix-vector equation is proposed. In addition, concrete solving methods are provided. Based on this, the solvability of the matrix equation AX=B with semi-tensor product is studied, and several examples are presented to illustrate the efficiency of the results.