Let  be the  matrix obtained by applying Φ entrywise to the transposed matrix where Φ is a nonstandard involution of the quaternion algebra  and  is a quaternion matrix. A square quaternion matrix A is said to be Φ-Hermitian if  where Φ is a nonstandard involution. In this paper, we consider the following quaternion matrix equation involving Φ-Hermicity  where A and B are given quaternion matrices. We derive some necessary and sufficient conditions for the existence of a Φ-Hermitian solution to this quaternion matrix equation in terms of the ranks and Moore-Penrose inverses of the coefficient matrices. We also give an expression of the general Φ-Hermitian solution to this quaternion matrix equation 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.