Least squares η-bi-Hermitian problems of the quaternion matrix equation (AXB,CXD) = (E,F)

For any and denote . Let , where is the th column of the identity matrix of order If and is called an -bi-Hermitian matrix. If and is called an -anti-bi-Hermitian matrix. -bi-Hermitian matrix set and -anti-bi-Hermitian matrix set are denoted by and , respectively. In this paper, the least squares problems of quaternion matrix equation over and are considered. The expressions of the least squares -bi-Hermitian solution with the least norm and the least squares -anti-bi-Hermitian solution with the least norm of the quaternion matrix equation are derived, respectively.