摘要
约束矩阵方程广泛应用于系统工程、自动控制、统计学、经济学、网络规划、
土木工程、振动理论等。本篇博士论文主要研究了以下几类约束矩阵方程问题以
及数值解法：
问题Ⅰ 给定矩阵A，B∈Rm×n，集合S(?)Rn×n，寻求X∈S，使得
‖AX-B‖=min．
问题Ⅱ 给定矩阵X，B∈Rm×n，集合S(?)Rm×m，寻求A∈S，使得
AX=B．
问题Ⅲ 给定特征值矩阵A∈Rk×k，A为对角阵或二阶块对角矩阵，以及相
应的特征向量矩阵X∈Rn×k，集合S(?)Rn×n，寻求A∈S，使得
AX=XA．
或者
‖AX-XA‖=min．
问题Ⅳ 给定矩阵A11∈Rm1×n1，A12∈Rm1×n2，A21∈Rm2×n1，m1+m2=
n1+n2=n，以及矩阵集合S(?)Rn×n，寻求子块A22∈Rm2×n2，使得完全化的矩阵
问题Ⅴ 给定A*∈Rn×n，设SE为上述问题的解集合，寻求解矩阵A∈SE，
使得
本文的主要研究成果如下：
1． 本文研究了问题Ⅰ在闭凸锥上的一种新的数值解法。创造性地利用闭凸锥上
的逼近理论、凸分析理论研究了最小二乘解的特征，结合最优化理论，提出了
投影梯度算法，理论上证明了算法的全局收敛性和线性收敛性。对8种常见
的闭凸锥，系统地提供了MATLAB程序，使求解变得方便、容易。
2． 对于问题Ⅱ，首次研究了约束矩阵集合S分别为广义反射矩阵、反对称正交
矩阵、部分等距算子、正交投影算子的情况下矩阵反问题的解，克服了约束矩
阵集合均为有界闭集带来的困难，成功地得到了有解的条件，并研究了最小
二乘解，提供了算法、部分MATLAB程序以及相应的数值实例。
求解几类特殊的约束矩阵方程的理论与算法研究
3.对于问题Hl，我们研究了Hamilton矩阵约束下矩阵逆特征值问题的最小二
乘解，首次给出了MATIAB程序计算最小二乘解和最小范数解;研究了正交
矩阵约束下逆特征值问题有解的条件，和最佳逼近解的求法，给出相应的算法
和数值实例。
4.对于问题W，我们继续研究了可逆矩阵的完全化问题，首次得到了通解、
最小范数解和最佳逼近解;首次研究反对称可逆矩阵完全化约束下矩阵的最
佳逼近间题，提供了算法计算唯一最佳逼近解;首次提出并研究了正交投影
算子的完全化问题，得到了有解的条件，并首次与矩阵的秩联合起来考虑完全
化，成功地编制了M ATLAB程序计算具有任意给定秩的解。
5.我们继续研究了有界闭集、子空间和线性流形上的最佳逼近问题，给出了求
解的方法和数值算例。
本篇博士论文得到了国家自然科学基金的资助。
本篇博士论文用拌玫江2:软件打印.
关键词:闭凸锥;投影梯度法;最小二乘解;矩阵反问题;矩阵逆特征值问题;矩
阵完全化;最佳逼近.
了
The constrained matrix equations have been widely applied in system engineering, autocontrol theory, economics, network programming, civil struction engineering and vibratory theory. The problems we will mainly discuss in the Ph.D. Thesis are as follows:Problem I Given two matrices A,B ∈ Rm×n, a constrained set S Rn×n. Find X ∈ S such thatAX - B = min.Problem II Given two matrices X, B 6 Rm×n, a constrained set S Rm×m. Find A e S such thatAX=B.Problem III Given an eigenvalue matrix ∈ Rk×k which is diagonal or 2 x 2 block diagonal, and the corresponding eigenvector matrix X ∈ Rn×k a constrained set S Rn×n. Find A ∈ S such thatAX = XAorProblem IV Given blocks , and a constrained set S Rn×n. Find a block A22 ∈ Gm2×m2 such thatProblem IV Given A* ∈ Rn×n, denote SE by the solution set of the above problems respectively. Find a solution matrix A ∈ SE such thatThe main fruits of the paper are as below: The numerical solutions for Problem I on the closed convex cone are sys-temically studied in the paper. Applying the approximation theory of closed convex cone and convex analysis in the creative way, the properties of least-squares solution are obtained. Using optimal theory, a gradient projection iteration is proposed to compute approximately the solutions of Problem I on
a closed convex cone. The global convergence and linear convergence have been theoritically proved. For eight familiar closed convex cones, a series of MATLAB procedures are provide to compute the solutions for them easily and conveniently.2. We discuss firstly the inverse problems (i.e. Problem II) of generalized reflection matrices, skew-symmetric orthogonal matrices, partial isometries and orthogonal projectors respectively, have obtained the necessary and sufficient conditions for solution existence. Furthermore, we also consider the least-sqaures solutions for them, provide algorithms or MATLAB functions to compute the solutions on PC.3. As for Problem III, we research the inverse eigenvalue problem of Hamiltonian matrices, get the methods to compute the least-squares solution and the best approximate solution for it, provide MATLAB procedures to calculate the least-squares solution and the least-squares solution with the least norm, we also investigate the inverse eigenvalue problem of orthogonal matrices, present algorithms to compute one solution when there exists at least a solution, to compute the least-squares solutions and optimal approximate solution. Some numerical examples are provided to illustrate the theory and the algorithms.4. Problem IV is matrix completion problem. We look into the matrix completion problem of nonsingular matrices, skew-symmetric invertible matrices and orthogonal projectors, deduce firstly the necessary and sufficient conditions for solution existence, provide some MATLAB functions or algorithms to compute the solutions, least-squares solutions and optimal approximate solutions.5. We also study the approximation problem(i.e. Problem V) on the subspace, the closed bounded set and linear manifold, give algorithms and corresponding numerical examples.This Ph.D. Thesis is supported by the National Natural Science Foundation of China.This Ph.D. Thesis is typeset by LATEX2ε
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