Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product

Abstract

This paper investigates the Sylvester-transpose matrix equation A⋉X+XT⋉B=C, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation A⋉X=C, the equation X⋉B=C, and the classical Sylvester-transpose matrix equation.