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.


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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