In this paper, several Kaczmarz-type numerical methods for solving the matrix equation $AX=B$ and $XA=C$ are proposed, where the coefficient matrix $A$ may be full rank or rank deficient. These methods are iterative methods without matrix multiplication. Theoretically, the convergence of these methods is proved. The numerical results show that these methods are more efficient than iterative methods involving matrix multiplication for high-dimensional matrices.

翻译：本文提出了求解矩阵方程$AX=B$和$XA=C$的若干Kaczmarz类数值方法，其中系数矩阵$A$可以是满秩或秩亏缺的。这些方法是不涉及矩阵乘法的迭代方法。理论上，我们证明了这些方法的收敛性。数值结果表明，对于高维矩阵，这些方法比涉及矩阵乘法的迭代方法更高效。