THE GENERAL φ-HERMITIAN SOLUTION TO MIXED PAIRS OF QUATERNION MATRIX SYLVESTER EQUATIONS

Let H-mxn be the space of m x n matrices over H, where H is the real quaternion algebra. Let A phi be 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 In this paper, some properties of the Moore-Penrose inverse of the quaternion matrix A phi are given. Two systems of mixed pairs of quaternion matrix Sylvester equations A(1)X - YB1 = Cl, A(2)Z - YB2 = C-2 and A(1)X - YB1 = C-1, A(2)Y - ZB(2) = C-2 are considered, where Z is phi-Hermitian. Some practical necessary and sufficient conditions for the existence of a solution (X, Y, Z) to those systems in terms of the ranks and Moore-Penrose inverses of the given coefficient matrices are presented. Moreover, the general solutions to these systems are explicitly given when they are solvable. Some numerical examples are provided to illustrate the main results.