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 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)].


15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A03 Vector spaces, linear dependence, rank, lineability
15A09 Theory of matrix inversion and generalized inverses
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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