The classical system of matrix equations , , where , and is a set of matrices over the quaternion algebra , is considered. After partitioning a solution of this system into block form matrices and with and the formulas of extreme ranks of the matrices are given.
Then, after characterizing the structure of the solutions , necessary and sufficient conditions for the uniqueness of the submatrices are established and the independence of the submatrices is analyzed. As applications the maximal and minimal ranks of the submatrices of the common inner inverse , partitioned into block form, of quaternion matrices and are presented. The properties of these matrices 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)].