Linear coupled matrix equations are widely utilized in applications, including stability analysis of control systems and robust control. In this paper, we establish the necessary and sufficient conditions for the consistency of the system of linear coupled matrix equations and derive an expression of the corresponding general solution (where it is solvable) over quaternion. Additionally, we investigate the necessary and sufficient conditions for the system of linear coupled matrix equations with construct to have a solution and derive a formula of its general solution (where it is solvable). Finally, an algorithm and an example were provided in order to further illustrate the primary outcomes of this paper.

线性耦合矩阵方程广泛应用于控制系统的稳定性分析和鲁棒控制等应用中。在本文中，我们建立了线性耦合矩阵方程组一致性的充要条件，并推导了四元数上相应通解（可解）的表达式。此外，我们还研究了具有构造的线性耦合矩阵方程组有解的充要条件，并推导了其通解（可解）的公式。最后，提供了算法和示例，以进一步说明本文的主要成果。