Solving the Generalized Sylvester Matrix Equation Σi=1pAiXBi + Σi=1q CjYDj = E Over Reflexive and Anti-reflexive Matrices

A matrix P is an element of R-nxn is called a generalized reflection if P-T = P and P-2=I. An nxn matrix A is said to be a reflexive (anti-reflexive) with respect to P if A = PAP (A = -PAP). In the present paper, two iterative methods are derived for solving the generalized Sylvester matrix equation Sigma(p)(i=1)A(i)XB(i) + Sigma(q)(i=1) CjYDj = E, (including the Sylvester and Lyapunov matrix equations as special cases) over reflexive and anti-reflexive matrices respectively. It is proven that the iterative methods, respectively, consistently converge to the reflexive and anti-reflexive solutions of the matrix equation for any initial reflexive and anti-reflexive matrices. Finally, a numerical example is given to demonstrate the effectiveness of the derived methods.