The block Kaczmarz method and its variants are designed for solving the over-determined linear system. They involve iteratively projecting the current point onto the solution space of a subset of constraints. In this work, by alternately dealing with two subproblems (i.e., linear system with multiple right-hand sides) using the block Kaczmarz method, we propose the {\it Alternating Randomized Block Kaczmarz} (ARBK) method to solve the linear matrix equation $AXB=F$, which incorporates a randomized index selection scheme to determine the subset of constraints. The convergence analysis reveals that the ARBK method has a linear convergence rate bounded by an explicit expression. Several numerical studies have been conducted to validate the theoretical findings.

翻译：块Kaczmarz方法及其变体用于求解超定线性方程组。这些方法通过迭代地将当前点投影到部分约束的解空间上进行求解。本文通过交替使用块Kaczmarz方法处理两个子问题（即多右端线性方程组），提出了交替随机块Kaczmarz（Alternating Randomized Block Kaczmarz, ARBK）方法来求解线性矩阵方程$AXB=F$。该方法采用随机索引选择方案来确定约束子集。收敛性分析表明，ARBK方法具有由显式表达式界定的线性收敛速率。本文通过多项数值实验验证了理论结果。