In this paper, based on an optimization problem, a sketch-and-project method for solving the linear matrix equation AXB = C is proposed. We provide a thorough convergence analysis for the new method and derive a lower bound on the convergence rate and some convergence conditions including the case that the coefficient matrix is rank deficient. By varying three parameters in the new method and convergence theorems, the new method recovers an array of well-known algorithms and their convergence results. Meanwhile, with the use of Gaussian sampling, we can obtain the Gaussian global randomized Kaczmarz (GaussGRK) method which shows some advantages in solving the matrix equation AXB = C. Finally, numerical experiments are given to illustrate the effectiveness of recovered methods.

翻译：本文基于一个优化问题，提出了一种求解线性矩阵方程AXB=C的素描-投影方法。我们对该新方法进行了全面的收敛性分析，推导了收敛速度的下界以及若干收敛条件（包括系数矩阵秩亏损的情形）。通过调整新方法中的三个参数及收敛定理，该方法可还原出一系列已知算法及其收敛结果。同时，采用高斯采样可得到高斯全局随机卡奇马兹（GaussGRK）方法，该方法在求解矩阵方程AXB=C方面展现出一定优势。最后，通过数值实验验证了所还原方法的有效性。