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

By Mehdi Dehghan, Masoud Hajarian. 2012

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Let P and Q be two generalized reflection matrices, i.e, P=PH, P2=I and Q=QH, Q2=I. An n×n matrix A is said to be generalized reflexive (generalized anti-reflexive) with respect to the matrix pair (P;Q) if A=PAQ (A=-PAQ). It is obvious that any n×m matrix is also a generalized reflexive with respect to the matrix pair (In;Im). By extending the conjugate gradient least square (CGLS) approach, the present paper treats two iterative algorithms to solve the system of matrix equations F1(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,…,Fm are the linear operators from Cn×n onto Cri×si and Ai∈Cri×si for i=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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/7dde09798620533f0ba54cbe4d51941a36e0ae5b