# Iterative algorithm for the generalized ‎$‎(P‎,‎Q)‎$‎-reflexive solution of a‎ ‎quaternion matrix equation with ‎$‎j‎$‎-conjugate of the unknowns

Document Type: Research Paper

Author

Shandong University of Finance and Economics

Abstract

In the present paper‎, ‎we propose an iterative algorithm for‎
‎solving the generalized $(P,Q)$-reflexive solution of the quaternion matrix‎
‎equation $\overset{u}{\underset{l=1}{\sum}}A_{l}XB_{l}+\overset{v} ‎{\underset{s=1}{\sum}}C_{s}\widetilde{X}D_{s}=F$‎. ‎By this iterative algorithm‎,
‎the solvability of the problem can be determined automatically‎. ‎When the‎
‎matrix equation is consistent over a generalized $(P,Q)$-reflexive matrix $X$‎, ‎a‎
‎generalized $(P,Q)$-reflexive solution can be obtained within finite iteration‎
‎steps in the absence of roundoff errors‎, ‎and the least Frobenius norm‎
‎generalized $(P,Q)$-reflexive solution can be obtained by choosing an‎
‎appropriate initial iterative matrix‎. ‎Furthermore‎, ‎the optimal approximate‎
‎generalized $(P,Q)$-reflexive solution to a given matrix $X_{0}$ can be‎
‎derived by finding the least Frobenius norm generalized $(P,Q)$-reflexive‎
‎solution of a new corresponding quaternion matrix equation‎. ‎Finally‎, ‎two‎
‎numerical examples are given to illustrate the efficiency of the proposed methods‎.

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### History

• Receive Date: 13 January 2013
• Revise Date: 06 August 2013
• Accept Date: 06 August 2013