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The randomized Kaczmarz algorithm is one of the most popular approaches forsolving large-scale linear systems due to its simplicity and efficiency. Inthis paper, we propose two classes of global randomized Kaczmarz methods forsolving large-scale linear matrix equations $AXB=C$, the global randomizedblock Kaczmarz algorithm and global randomized average block Kaczmarzalgorithm. The feature of global randomized block Kaczmarz algorithm is thefact that the current iterate is projected onto the solution space of thesketched matrix equation at each iteration, while the global randomized averageblock Kaczmarz approach is pseudoinverse-free and therefore can be deployed onparallel computing units to achieve significant improvements in thecomputational time. We prove that these two methods linearly converge in themean square to the minimum norm solution $X_*=A^\dag CB^\dag$ of a given linearmatrix equation. The convergence rates depend on the geometric properties ofthe data matrices and their submatrices and on the size of the blocks.Numerical results reveal that our proposed algorithms are efficient andeffective for solving large-scale matrix equations. In particular, they canalso achieve satisfying performance when applied to image deblurring problems.

On global randomized block Kaczmarz algorithm for solving large-scale matrix equations