Solvability conditions and general solution for mixed Sylvester equations

Some Results on Characterizations of Matrix Partial Orderings

“…It is well known that rank of matrix is an important tool in matrix theory and its applications, and many problems are closely related with the ranks of some matrix expressions under some restrictions (see [11][12][13][14][15] for details). Our aim in this paper is to characterize the left-star, right-star, star, and sharp partial orderings by applying rank equalities.…”

Biconjugate residual algorithm for solving general Sylvester-transpose matrix equations

“…appears in the model reduction and stability, reachability, observability and controllability analysis of discrete-time and continuous-time linear systems [36,[38][39][40]. During the last two decades, the linear matrix equations have drawn much attention due to their wide applications [3,14,27,28,35,42]. Li and Wang introduced the weighted steepest descent algorithms to solve the general linear matrix equation including the Lyapunov and Sylvester matrix equations [31].…”

Solvability conditions for mixed sylvester equations in rings

“…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].…”

“…Liu [18] also gave a solvability condition to (1). Wang and He [31] presented new necessary and sufficient solvability conditions for the system (1), and gave an expression of the general solution when it is solvable. When X = Z, the system (1) becomes pairs of generalized Sylvester equations…”