This paper presents an efficient method for obtaining the least squares Hermitian solutions of the reduced biquaternion matrix equation $(AXB, CXD) = (E, F )$. The method leverages the real representation of reduced biquaternion matrices. Furthermore, we establish the necessary and sufficient conditions for the existence and uniqueness of the Hermitian solution, along with a general expression for it. Notably, this approach differs from the one previously developed by Yuan et al. $(2020)$, which relied on the complex representation of reduced biquaternion matrices. In contrast, our method exclusively employs real matrices and utilizes real arithmetic operations, resulting in enhanced efficiency. We also apply our developed framework to find the Hermitian solutions for the complex matrix equation $(AXB, CXD) = (E, F )$, expanding its utility in addressing inverse problems. Specifically, we investigate its effectiveness in addressing partially described inverse eigenvalue problems. Finally, we provide numerical examples to demonstrate the effectiveness of our method and its superiority over the existing approach.

翻译：本文提出了一种高效方法，用于求解约化四元数矩阵方程 $(AXB, CXD) = (E, F)$ 的最小二乘Hermitian解。该方法利用了约化四元数矩阵的实表示。此外，我们建立了Hermitian解存在且唯一的充分必要条件，并给出了其一般表达式。值得注意的是，该方法与Yuan等人(2020)之前基于约化四元数矩阵复表示的方法不同。相比之下，我们的方法仅使用实矩阵和实算术运算，从而提高了效率。我们还将所提出的框架应用于求解复矩阵方程 $(AXB, CXD) = (E, F)$ 的Hermitian解，扩展了其在处理逆问题中的实用性。具体而言，我们探究了其在部分描述逆特征值问题中的有效性。最后，我们提供了数值算例，以证明所提方法的有效性及其相较于现有方法的优越性。