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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.

MSC:

15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A09 Theory of matrix inversion and generalized inverses
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