We put forward and analyze in details an iterative method to find the mixed solutions of a matrix equation with sub-matrix constraints. The convergence of the approximated solution sequence generated by the iterative method is investigated, showing that if the constrained matrix equation is consistent, the mixed solution group can be obtained after a finite number of iterations. Moreover, for a given matrix, its best approximation is obtained, which is the mixed solution of the matrix equation with sub-matrix constraints. Finally, a large number of numerical experiments are carried out, and results show that the algorithm is effective not only in image restoration, but also in the general case for both small-scale and large-scale matrices.

提出并详细分析了求解带子矩阵约束的矩阵方程混合解的迭代方法。考察了迭代法生成的近似解序列的收敛性，表明如果约束矩阵方程一致，经过有限次迭代就可以得到混合解群。而且，对于给定的矩阵，得到了它的最佳近似，即带有子矩阵约束的矩阵方程的混合解。最后，进行了大量的数值实验，结果表明该算法不仅在图像恢复方面有效，而且在小尺度和大尺度矩阵的一般情况下都是有效的。