Let ${\eta \mathbb{H}\mathbb{Q}}^{k\times k}$ and ${\eta \mathbb{A}\mathbb{Q}}^{k\times k}$ represent the sets of all $k\times k$ η-Hermitian quaternion matrices and η-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 equation AXB + CY D = E over $X\in {\eta \mathbb{H}\mathbb{Q}}^{n\times n},Y\in {\eta \mathbb{A}\mathbb{Q}}^{k\times k}$ into 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.

让 ${\eta \mathbb{H}\mathbb{问}}^{克\times 克}$ 和 ${\eta \mathbb{一种}\mathbb{问}}^{克\times 克}$ 代表所有的集合 $克\times 克$ η -Hermitian 四元数矩阵和η -anti-Hermitian 四元数矩阵，分别。上的四元数矩阵和它的特殊结构的真实表示矩阵的基础上，我们转换四元数矩阵方程的最小二乘问题AXB  +  CY d  =  È过$X\in {\eta \mathbb{H}\mathbb{问}}^{n\times n},是\in {\eta \mathbb{一种}\mathbb{问}}^{克\times 克}$进入自由变量上的实矩阵方程的相应问题，然后我们建立其唯一的最小范数最小二乘解。我们得到的表达式仅由实数矩阵表示，并且算法仅包含实数运算。因此，它们非常简单和方便。与现有方法[SF Yuan, QW Wang, and X. Zhang, Least-squares problem for the Quaternion matrix equation AXB + CYD = E over different constrained matrix , Int. J. 计算。数学。90 (2013), pp. 565–576]，最后两个例子表明我们的方法更有效、更优越。