Let  be the set of matrices over the quaternion algebra Let  be the  matrix obtained by applying Φ entrywise to the transposed matrix  where  and Φ is a nonstandard involution of  is said to be Φ-Hermitian if  where Φ is a nonstandard involution. In this paper, we consider the following system of quaternion matrix equations  where A, B, C, and D are given quaternion matrices, the variable X is Φ-Hermitian. We give some necessary and sufficient conditions for the existence of a Φ-Hermitian solution to this system in terms of the ranks and Moore-Penrose inverses of the coefficient matrices. We also present an expression of the general Φ-Hermitian solution to this system when it is solvable. We also provide a numerical example to illustrate the main result.

quaternion, matrix equation, Moore-Penrose inverse, involution, Φ‑Hermitian solution.