The least squares η-Hermitian problems of quaternion matrix equation AHXA + BHYB=C

For any A=A1+A2j?Qnxn and ?? {i,j,k} denote A?H = -?AH?. If A?H = A,A is called an ?-Hermitian matrix. If A?H =-A,A is called an ?-anti-Hermitian matrix. Denote ?-Hermitian matrices and ?-anti-Hermitian matrices by ?HQnxn and ?AQnxn, respectively. In this paper, we consider the least squares ?-Hermitian problems of quaternion matrix equation AHXA+ BHYB = C by using the complex representation of quaternion matrices, the Moore-Penrose generalized inverse and the Kronecker product of matrices. We derive the expressions of the least squares solution with the least norm of quaternion matrix equation AHXA + BHYB = C over [X,Y] ? ?HQnxn x ?HQkxk, [X,Y] ? ?AQnxn x ?AQkxk, and [X,Y] ? ?HQnxn x ?AQkxk, respectively.