Abstract
<abstract><p>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.</p></abstract>
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