Real Representation Approach to Quaternion Matrix Equation Involving φ-Hermicity

For a quaternion matrix A, we denote by A phi the matrix obtained by applying phi entrywise to the transposed matrix AT, where phi is a nonstandard involution of quaternions. A is said to be phi-Hermitian or phi-skew-Hermitian if A=A phi or A=-A phi, respectively. In this paper, we give a complete characterization of the nonstandard involutions phi of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a phi-Hermitian solution or phi-skew-Hermitian solution to the quaternion matrix equation AX=B. Moreover, we give solutions of the quaternion equation when it is solvable.