In this paper, the octonion matrix equation \(AXB=C\) is studied based on semi-tensor product of matrices. Firstly, we propose the left real element representation and the right real element representation of octonion. Then we obtain the expression of the least squares Hermitian solution to the octonion matrix equation \(AXB=C\) by combining these representations with \(\mathcal {H}\)-representation of the special matrices. In addition, we also put forward the equivalent condition of existence and general expression of the Hermitian solution to the octonion matrix equation \(AXB=C.\) Finally, the validity and stability of our method is verified by numerical experiments.

本文基于矩阵的半张量积研究了八元数矩阵方程\(AXB=C\) 。首先，我们提出八元数的左实元表示和右实元表示。然后，我们通过将这些表示与特殊矩阵的表示相结合，获得八元数矩阵方程\(AXB=C\)的最小二乘埃尔米特解的表达式。此外，我们还提出了八元数矩阵方程\(AXB=C\)埃尔米特解的等价存在条件和一般表达式。最后，通过数值实验验证了该方法的有效性和稳定性。