“…Many papers have presented different approaches for several matrix equations [7-9, 12-14, 17, 19, 20]. Especially, many problems in control theory can be transformed into the Sylvester matrix equations, such as singular system control [4,21], robust control [3,26], neural network [25,36]. The solvability of linear equations is a fundamental problem, and various results are developed, such as solvability conditions of linear equations for matrices over the complex field [1,2,10,11,18,22,23,[29][30][31][32][33][34]37], solvability conditions of linear equations over algebras or rings [5,6,24,27,28,35].…”