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Zbl 1222.15022
Zhang, Qin; Wang, Qing-Wen
The $(P,Q)$-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations. (English)
[J] Appl. Math. Comput. 217, No. 22, 9286-9296 (2011). ISSN 0096-3003

The paper deals with extremal rank solutions to a system of quaternion matrix equations. $H^{m \times n}$ denotes the set of $m \times n$ matrices over the real quaternion algebra $$H=\{a_0+a_1i+a_2j+a_3k \ : \ i^2=j^2=k^2=ijk=-1 \ \text {and} \ a_0,a_1,a_2,a_3 \ \text {are real numbers} \}.$$ A matrix $A \in H^{m \times n}$ is called $(P,Q)$-symmetric (or $(P,Q)$-skewsymmetric) if $A=PAQ$ (or $A=-PAQ$), where $P \in H^{m \times m}$ and $Q \in H^{n \times n}$ are involution matrices. Consider the system of matrix equations over $H$ $$ AX=B, \ XC=D. \tag*$$ In this work, the authors analyze the $(P,Q)$-(skew)symmetric maximal and minimal rank solutions of this system. They obtain necessary and sufficient conditions for the existence of $(P,Q)$-symmetric and $(P,Q)$-skewsymmetric solutions to the above system and give the expressions of such solutions when the solvability conditions are satisfied. The authors also establish formulas of maximal and minimal ranks of $(P,Q)$-symmetric and $(P,Q)$-skewsymmetric solutions of (*) and derive the expressions of $(P,Q)$-(skew)symmetric maximal and minimal rank solutions of (*). Finally, the authors present a numerical example that confirms the theoretical results obtained.
[Juan Ramon Torregrosa Sanchez (Valencia)]

MSC 2000:: *15A24 Matrix equations
15A33 Matrices over special rings
15A09 Matrix inversion

Keywords: Moore-Penrose inverse; $(P,Q)$-symmetric matrix; $(P,Q)$-skewsymmetric matrix; minimal rank; maximal rank; extremal rank solutions; system of quaternion matrix equations; quaternion algebra; numerical example

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