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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 XB 1 =C 1 , A 2 XB 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 m×n , is considered. After partitioning a solution X of this system into 2×2 block form matrices X 1 k 1 ×l 1 ,X 2 k 1 ×l 2 ,X 3 k 2 ×l 1 and X 4 k 2 ×l 2 (k 1 +k 2 =k,l 1 +l 2 =l) with A 1 ,A 2 m×k ,B 1 ,B 2 l×n and C 1 ,C 2 m×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 2×2 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:
15A24Matrix equations and identities
15A33Matrices over special rings
15A03Vector spaces, linear dependence, rank
15A09Matrix inversion, generalized inverses
11R52Quaternion and other division algebras: arithmetic, zeta functions