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Zbl 1227.15015
Lian, Dezhong; Wang, Qingwen; Tang, Yan
Extreme ranks of a partial banded block quaternion matrix expression subject to some matrix equations with applications. (English)
[J] Algebra Colloq. 18, No. 2, 333-346 (2011). ISSN 1005-3867

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.
[Huajun Huang (Auburn)]

MSC 2000:: *15A24 Matrix equations
15A33 Matrices over special rings
15A03 Vector spaces
15A09 Matrix inversion

Keywords: quaternion matrix equation; maximal rank; minimal rank; quadratic matrix expression; quaternion algebra

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