Let H be the real quaternion algebra and Hmxn denote the set of all m x n
matrices over H. For A ? Hm x n, we denote by A? the n x m matrix obtained by
applying ? entrywise to the transposed matrix At, where ? is a nonstandard
involution of H. A ? Hnxn is said to be ?-Hermitian if A = A?. In this
paper, we construct a simultaneous decomposition of four real quaternion
matrices with the same row number (A,B,C,D), where A is ?-Hermitian, and
B,C,D are general matrices. Using this simultaneous matrix decomposition, we
derive necessary and sufficient conditions for the existence of a solution to
some real quaternion matrix equations involving ?-Hermicity in terms of
ranks of the given real quaternion matrices. We also present the general
solutions to these real quaternion matrix equations when they are solvable.
Finally some numerical examples are presented to illustrate the results of
this paper.