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Ranks of submatrices in a general solution to a quaternion system with applications. (English) Zbl 1229.15017

The classical system of matrix equations ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$, where ${A}_{1},{B}_{1},{C}_{1}$, ${A}_{2},{B}_{2},{C}_{2}$ and $X$ is a set of matrices over the quaternion algebra ${ℍ}^{m×n}$, is considered. After partitioning a solution $X$ of this system into $2×2$ block form matrices ${X}_{1}\in {ℍ}^{{k}_{1}×{l}_{1}},{X}_{2}\in {ℍ}^{{k}_{1}×{l}_{2}},{X}_{3}\in {ℍ}^{{k}_{2}×{l}_{1}}$ and ${X}_{4}\in {ℍ}^{{k}_{2}×{l}_{2}}$ $\left({k}_{1}+{k}_{2}=k,{l}_{1}+{l}_{2}=l\right)$ with ${A}_{1},{A}_{2}\in {ℍ}^{m×k},{B}_{1},{B}_{2}\in {ℍ}^{l×n}$ and ${C}_{1},{C}_{2}\in {ℍ}^{m×n}$ the formulas of extreme ranks of the matrices ${X}_{i}$ $\left(i=1,2,3,4\right)$ are given.

Then, after characterizing the structure of the solutions ${X}_{i}$, necessary and sufficient conditions for the uniqueness of the submatrices ${X}_{i}$ are established and the independence of the submatrices ${X}_{i}$ is analyzed. As applications the maximal and minimal ranks of the submatrices of the common inner inverse $G$, partitioned into $2×2$ block form, of quaternion matrices $M$ and $N$ are presented. The properties of these matrices $G$ are also described.

This paper represents the generalization of results given by Y. Tian [J. Franklin Inst. 346, No. 6, 557–569 (2009; Zbl 1168.15307)] and Y. Liu [J. Appl. Math. Comput. 31, No. 1–2, 71–80 (2009; Zbl 1186.15013)].

##### MSC:
 15A24 Matrix equations and identities 15A33 Matrices over special rings 15A03 Vector spaces, linear dependence, rank 15A09 Matrix inversion, generalized inverses 11R52 Quaternion and other division algebras: arithmetic, zeta functions