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Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product
Author(s)
Jaiprasert, Janthip
Date Issued
June 1, 2022
Type
Article
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.
Citation
Symmetry, 14(6), 2022