Solutions to matrix equations X−AXB=CY+R and X−AX̂B=CY+R

The present work proposed an alternative approach to find the closed-form solutions of the nonhomogeneous Yakubovich matrix equation X−AXB=CY+R. Based on the derived closed-form solution to the nonhomogeneous Yakubovich matrix equation, the solutions to the nonhomogeneous Yakubovich quaternion j-conjugate matrix equation X−AX̂B=CY+R are obtained by the use of the real representation of a quaternion matrix. The existing complex representation method requires the coefficient matrix A to be a block diagonal matrix over complex field. In contrast in this publication we allow a quaternion matrix of any dimension. As an application, eigenstructure assignment problem for descriptor linear systems is considered.