Solving the generalized sylvester matrix equation Σ i=1 p AiXBi + Σ j=1 q CjYDj=E over reflexive and anti-reflexive matrices

Abstract

A matrix P ∈ ℝ n×n is called a generalized reflection if P T = P and P 2=I. An n×n 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 Σ =1 A i XB i + Σ =1 C j YD j =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.