Solvability of the Sylvester equation AX−XB=C under left semi-tensor product

Abstract

This paper investigates the solvability of the Sylvester matrix equation AX−XB=C with respect to left semi-tensor product. Firstly, we discuss the matrix-vector equation AX−XB=C under semi-tensor product. A necessary and sufficient condition for the solvability of the matrix-vector equation and specific solving methods are studied and given. Based on this, the solvability of the matrix equation AX−XB=C under left semi-tensor product is discussed. Finally, several examples are presented to illustrate the efficiency of the results.

Keywords

• sylvester matrix equation
• left semi-tensor product
• solvability
• matrix-vector equation
• necessary and sufficient condition