GENERAL φ-HERMITIAN SOLUTION TO A SYSTEM OF QUATERNION MATRIX EQUATIONS

Let H-mxn be the set of m x n matrices over the quaternion algebra H. Let A(phi) the n x m matrix obtained by applying phi entrywise to the transposed matrix A(T), where A is an element of H-mxn and phi is a nonstandard involution of H. A is an element of H-mxn is said to be phi-Hermitian if A = A(phi) , where phi is a nonstandard involution. In this paper, we consider the following system of quaternion matrix equations AX = C, XB = D, where A, B, C, and D are given quaternion matrices, the variable X is phi-Hermitian. We give some necessary and sufficient conditions for the existence of a phi-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 phi-Hermitian solution to this system when it is solvable. We also provide a numerical example to illustrate the main result.