Solving the Generalized Sylvester Matrix Equation ∑^p_i=1 A_iXB_i + ∑_q_j=1 C_jYD_j = E Over Reflexive and Anti-reflexive Matrices

다국어 초록 (Multilingual Abstract)

A matrix is P ∈ R^n×n 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 ∑^p_i=1 A_iXB_i + ∑_q_j=1 C_jYD_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, re-spectively, 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.

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