A simultaneous decomposition for seven matrices with applications

Let Hm×n be the set of all m×n matrices over the quaternion algebra H. In this paper, we construct a simultaneous decomposition for seven matrices with compatible sizes: A∈Hm×n,B∈Hm×p1,C∈Hm×p2,D∈Hm×p3,E∈Hq1×n,F∈Hq2×n and G∈Hq3×n. As applications of the simultaneous matrix decomposition, we give some solvability conditions, general solutions, as well as the range of ranks of the general solutions to the following two generalized Sylvester matrix equations BXE+CYF+DZG=A and BX+WE+CYF+DZG=A, where A,B,C,D,E,F, and G are given quaternion matrices. Moreover, we provide some numerical examples to illustrate the results of this paper.