In this paper, several row and column orthogonal projection methods are proposed for solving matrix equation $AXB=C$, where the matrix $A$ and $B$ are full rank or rank deficient and equation is consistent or not. These methods are iterative methods without matrix multiplication. It is theoretically proved these methods converge to the solution or least-squares solution of the matrix equation. Numerical results show that these methods are more efficient than iterative methods involving matrix multiplication for high-dimensional matrix.

翻译：本文针对矩阵方程$AXB=C$，提出了若干行列正交投影方法，其中矩阵$A$与$B$可以是满秩或秩亏损，方程可以相容或不相容。这些方法均为无需矩阵乘法的迭代方法。理论上证明了这些方法收敛于矩阵方程的解或最小二乘解。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为高效。