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

Abstract

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 .