An Efficient Real Representation Method For Least Squares Problem Of The Quaternion Constrained Matrix Equation Axb Plus Cy D = E

Letandrepresent the sets of all eta-Hermitian quaternion matrices and eta-anti-Hermitian quaternion matrices, respectively. On the basis of the real representation matrix of a quaternion matrix and its particular structure, we convert the least squares problem of the quaternion matrix equationAXB + CY D = Eoverinto the corresponding problem of the real matrix equation over free variables, and then we establish its unique minimal norm least squares solution. Our resulting expressions are expressed only by real matrices, and the algorithm only includes real operations. Consequently, they are very simple and convenient. Compared with the existing method [S.F. Yuan, Q.W. Wang, and X. Zhang,Least-squares problem for the quaternion matrix equation AXB + CYD = E over different constrained matrices, Int. J. Comput. Math. 90 (2013), pp. 565-576], the final two examples show that our method is more efficient and superior.