On the generalized reflexive and anti-reflexive solutions to a system of matrix equations

Let P and Q   be two generalized reflection matrices, i.e, P=PHP=PH, P2=IP2=I and Q=QHQ=QH, Q2=IQ2=I. An n×nn×n matrix A   is said to be generalized reflexive (generalized anti-reflexive) with respect to the matrix pair (P;Q)(P;Q) if A=PAQA=PAQ(A=-PAQ)(A=-PAQ). It is obvious that any n×mn×m matrix is also a generalized reflexive with respect to the matrix pair (In;Im)(In;Im). By extending the conjugate gradient least square (CGLS) approach, the present paper treats two iterative algorithms to solve the system of matrix equationsF1(X)=A1,F2(X)=A2,⋮⋮⋮Fm(X)=Am,(including the Sylvester and Lyapunov matrix equations as special cases) over the generalized reflexive and anti-reflexive matrices, where F1,F2,…,FmF1,F2,…,Fm are the linear operators from Cn×nCn×n onto Cri×siCri×si and Ai∈Cri×siAi∈Cri×si for i=1,2,…,mi=1,2,…,m. When this system is consistent over the generalized reflexive (generalized anti-reflexive) matrix, it is proved that the first (second) iterative algorithm converges to a generalized reflexive (generalized anti-reflexive) solution for any initial generalized reflexive (generalized anti-reflexive) matrix. Also the first (second) iterative algorithm can obtain the the least Frobenius norm generalized reflexive (generalized anti-reflexive) solution for special initial generalized reflexive (generalized anti-reflexive) matrix. Furthermore, the optimal approximation generalized reflexive (generalized anti-reflexive) solution to a given generalized reflexive (generalized anti-reflexive) matrix can be derived by finding the least Frobenius norm generalized reflexive (generalized anti-reflexive) solution of a new system of matrix equations. Finally, we test the proposed iterative algorithms and show their effectiveness using numerical examples.