Least-squares problem for the quaternion matrix equation AXB+CYD=E over different constrained matrices

For any A=A ₁+A ₂ j∈Q ^n×n and η∈<texlscub>i, j, k</texlscub>, denote A ^{η H} =?η A ^H η. If A ^{η H} =A, A is called an $\eta$-Hermitian matrix. If A ^{η H} =?A, A is called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices by η HQ ^n×n and η AQ ^n×n , respectively.

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 for the quaternion matrix equation AXB+CYD=E over X∈η HQ ^n×n and Y∈η AQ ^n×n .