In this paper, we apply the semi-tensor product of matrices and the real vector representation of quaternion matrix to solve quaternion matrix equations for the first time. We derive the expressions of the least squares Hermitian, Persymmetric and Bisymmetric solution for the quaternion matrix equation \(AXB=C\), respectively. Furthermore, we also put forward the equivalent conditions of existence and general expression of the Hermitian, Persymmetric and Bisymmetric solution to the quaternion matrix equation \(AXB=C\), respectively. Finally, the validity of our method is verified by numerical experiments with different dimensions.

中文翻译：

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四元数矩阵方程的几种特殊最小二乘解$$AXB=C$$AXB=C

在本文中，我们首次应用矩阵的半张量积和四元数矩阵的实向量表示来求解四元数矩阵方程。我们分别推导出四元数矩阵方程\(AXB=C\) 的最小二乘厄米、过对称和双对称解的表达式。此外，我们还分别提出了四元数矩阵方程\(AXB=C\)的厄米、过对称和双对称解的等价存在条件和一般表达式。最后，通过不同维度的数值实验验证了我们方法的有效性。