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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 $$\left[\matrix M_{11}&M_{12}&X\\ M_{21}&M_{22}&M_{23}\\ Y&M_{32}&M_{33}\endmatrix\right]$$ 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.

15A24Matrix equations and identities
15B33Matrices over special rings (quaternions, finite fields, etc.)
15A03Vector spaces, linear dependence, rank
15A09Matrix inversion, generalized inverses
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