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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 ${\mathbb H}^{m \times n}$, is considered. After partitioning a solution $X$ of this system into $2 \times 2$ block form matrices $X_1 \in {\mathbb H}^{k_1 \times l_1}, X_2 \in {\mathbb H}^{k_1 \times l_2}, X_3 \in {\mathbb H}^{k_2 \times l_1}$ and $ X_4 \in {\mathbb H}^{k_2 \times l_2}$ $(k_1 + k_2 = k, l_1 + l_2 =l)$ with $A_1, A_2 \in {\mathbb H}^{m \times k}, B_1, B_2 \in {\mathbb H}^{l \times n}$ and $C_1, C_2 \in {\mathbb H}^{m \times n}$ the formulas of extreme ranks of the matrices $X_i$ $(i = 1,2,3,4)$ 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 $2 \times 2$ 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 {\it Y. Tian} [J. Franklin Inst. 346, No. 6, 557--569 (2009; Zbl 1168.15307)] and {\it Y. Liu} [J. Appl. Math. Comput. 31, No. 1--2, 71--80 (2009; Zbl 1186.15013)].

15A24Matrix equations and identities
15B33Matrices over special rings (quaternions, finite fields, etc.)
15A03Vector spaces, linear dependence, rank
15A09Matrix inversion, generalized inverses
11R52Quaternion and other division algebras: arithmetic, zeta functions
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