Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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RANKS OF SUBMATRICES IN A GENERAL SOLUTION TO A QUATERNION SYSTEM WITH APPLICATIONS

  • Zhang, Hua-Sheng (School of Mathematics Science Liaocheng University) ;
  • Wang, Qing-Wen (Department of Mathematics Shanghai University, Division of Mathematical Sciences Nanyang Technological University)
  • Received : 2010.02.09
  • Published : 2011.09.30
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Abstract

Assume that X, partitioned into $2{\times}2$ block form, is a solution of the system of quaternion matrix equations $A_1XB_1$ = $C_1,A_2XB_2=C_2$. 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 $2{\times}2$ 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.

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