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The $(P,Q)$-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations. (English) Zbl 1222.15022
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.

15A24Matrix equations and identities
15B33Matrices over special rings (quaternions, finite fields, etc.)
15A09Matrix inversion, generalized inverses
Full Text: DOI
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