Ranks of submatrices in a general solution to a quaternion system with applications
Bull. Korean Math. Soc. 2011 Vol. 48, No. 5, 969-990
Printed September 1, 2011
Hua-Sheng Zhang and Qing-Wen Wang
Liaocheng University, Nanyang Technological University
Abstract : Assume that $X$, partitioned into $2\times2$ block form, is a solution of the system of quaternion matrix equations $A_{1} XB_{1}=C_{1},A_{2}XB_{2}=C_{2}.$ We in this paper give the maximal and minimal ranks of the submatrices in $X,$ and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. As applications, we consider the common inner inverse $G,$ partitioned into $2\times2$ block form, of two quaternion matrices $M$ and $N.$ We present the formulas of the maximal and minimal ranks of the submatrices of $G,$ and describe the properties of the submatrices of $G$ as well. The findings of this paper generalize some known results in the literature.
Keywords : matrix equation, minimal rank, maximal rank, generalized inverse, quaternion matrix, partitioned matrix
MSC numbers : 15A03, 15A09, 15A24, 11R52
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