On solutions to the matrix equations XB - AX = CY and XB - A(X)over-cap = CY

The solution of the generalized Sylvester real matrix equation XB - AX = CY is important in stability analysis and controller design in linear systems. This paper presents an explicit solution to the generalized Sylvester real matrix equation XB - AX = CY. Based on the derived explicit solution to the considered generalized Sylvester real matrix equation, a new approach is provided for obtaining the solutions to the generalized Sylvester quaternion j-conjugate matrix equation XB - A (X) over cap = CY using the real representation of a quaternion matrix. The closed form solution is established in an explicit form for this generalized Sylvester quaternion j-conjugate matrix equation. The existing complex representation method requires the coefficient matrix A to be a block diagonal matrix over complex field, while it is any suitable dimension quaternion matrix in the present paper. Therefore, we generalize the existing results. (C) 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.