# RANKS OF SUBMATRICES IN A GENERAL SOLUTION TO A QUATERNION SYSTEM WITH APPLICATIONS

• Zhang, Hua-Sheng ;
• Wang, Qing-Wen
• Published : 2011.09.30
• 42 13

#### 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.

#### Keywords

matrix equation;minimal rank;maximal rank;generalized inverse;quaternion matrix;partitioned matrix

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#### Cited by

1. Inertias and ranks of some Hermitian matrix functions with applications vol.10, pp.1, 2012, https://doi.org/10.2478/s11533-011-0117-9
2. The Optimization on Ranks and Inertias of a Quadratic Hermitian Matrix Function and Its Applications vol.2013, 2013, https://doi.org/10.1155/2013/961568