In this paper, we study the quaternion matrix equation (1.1) by using the semi-tensor product of matrices. First, we propose a real vector representation of a quaternion matrix and study its properties. Then combining this real vector representation with semi-tensor product of matrices, the explicit expressions of the least squares solution, the least squares J-self-conjugate solution and the least squares anti-J-self-conjugate solution of (1.1) are proposed. We give the equivalence conditions for the compatibility of (1.1) and the expressions of the minimal norm solution with above-mentioned properties. We also propose the corresponding algorithms and give numerical experiments to verify the effectiveness of our methods.

在本文中，我们利用矩阵的半张量积来研究四元数矩阵方程（1.1）。首先，我们提出了四元数矩阵的实向量表示并研究其性质。然后将此实向量表示与矩阵的半张量积相结合，提出了(1.1)的最小二乘解、最小二乘J-自共轭解和最小二乘反J-自共轭解的显式表达式. 我们给出了（1.1）的兼容性的等价条件和具有上述性质的最小范数解的表达式。我们还提出了相应的算法并给出了数值实验来验证我们方法的有效性。