多变量线性矩阵方程问题出现在数学、力学、线性系统和控制理论等众多学科领域，是数值代数的重要分支。交替投影类算法是数值求解相交集合交点的最有效方法之一，该算法目前已广泛地应用于逆特征值问题、协方差控制和图像复原等诸多领域。本项目将首次系统地应用该算法理论研究凸约束下多变量线性矩阵方程求解问题。具体研究内容为：1）结合最佳逼近理论，研究任意矩阵对（束）在所涉及的闭凸集和仿射子空间上投影矩阵的高效计算；2）建立求解问题的交替投影算法及其加速算法，并分析加速算法中各参数对算法收敛性和收敛速度的影响；3）利用算子理论分析加速算法的收敛性。本项目的研究将丰富和发展矩阵方程问题的研究，促进矩阵理论的应用和发展。

The multivariable linear matrix equation problem appears frequently in many fields, such as mathematics, mechanics, linear system and control theory and so on, and it is an important branch of the numerical algebra. Alternating projecting method can often be a very effective means of finding a point in the intersection of two or more convex sets, which has been widely used in inverse eigenvalue problem, covariance control design and image restoration and so on. In this project we will for the first time apply this algorithm and its variants to solve the convex constrained multivariable linear matrix equation problems. The detailed research contents are: 1) combining with optimal approximation theory, we characterize numerically the projection onto every one of the closed convex sets and affine subspaces involved in the algorithms; 2) we establish alternating projection method to solve the problems proposed in this project, and also study various accelerated versions of the alternating projection method, and then observe how the relaxation parameter and the step size influence both the convergence and the convergence rate of these accerated algorithms; 3) in addition, we analyze the convergence of these accelerate algorithms by operator theory. In conclusion, apart from enriching and developing the research of matrix equation, this project can be also capable of promoting the development and application of matrix theory.