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Extreme ranks of a partial banded block quaternion matrix expression subject to some matrix equations with applications. (English) Zbl 1227.15015

Authors’ abstract: We establish the formulas of the maximal and minimal ranks of a \(3\times3\) partial banded block matrix
\[ \left[\begin{matrix} M_{11}&M_{12}&X\\ M_{21}&M_{22}&M_{23}\\ Y&M_{32}&M_{33}\end{matrix}\right] \]
where \(X\) and \(Y\) are a pair of variant quaternion matrices subject to linear quaternion matrix equations \(A_1X=C_1\), \(XB_1=C_2\), \(A_2Y=D_1\), \(YB_2=D_2\). As applications, we present a necessary and sufficient condition for the solvability to the quadratic system \(A_1X=C_1\), \(XB_1=C_2\), \(A_2Y=D_1\), \(YB_2=D_2\), \(XPY=J\) over the quaternion algebra. We also give the conditions for the rank invariance of the quadratic matrix expression \(XPY=J\) subject to the linear quaternion matrix equations mentioned above.

MSC:

15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A03 Vector spaces, linear dependence, rank, lineability
15A09 Theory of matrix inversion and generalized inverses
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