We derive an iterative procedure for solving a generalized Sylvester matrix equation \(AXB+CXD = E\), where \(A,B,C,D,E\) are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation \(AXB=C\), the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

我们导出一个迭代过程来求解广义Sylvester矩阵方程\（AXB + CXD = E \），其中\（A，B，C，D，E \）是一致的矩形矩阵。我们的算法基于梯度和层次识别原理。我们将矩阵迭代过程转换为具有矩阵系数的一阶线性差分矢量方程。Banach收缩原理表明，当且仅当收敛因子属于开放区间时，近似解的序列才会收敛到任何初始矩阵的精确解。收缩原理还给出了收敛速度和误差分析，并由相关迭代矩阵的光谱半径控制。我们获得最快的收敛因子，从而使迭代矩阵的频谱半径最小。特别是，我们获得了矩阵方程\（AXB = C \）的迭代算法，Sylvester方程和Kalman-Yakubovich方程。我们给出了该算法的数值实验，以说明其适用性，有效性和效率。