Löwner偏序下矩阵方程的不等式约束解以及矩阵不等式的数值解的迭代算法

It is well known that the nonnegative definite solutions of matrix equations and matrix inequalities have wide applications in power system, multivariate linear model and control theory, however, only a little research work was done on the numerical iterative algorithms to these matrix equations and matrix inequalities. So, in this project, by modifying some classical matrix iterative algorithms, we will establish the numerical iterative algorithms for the nonnegative definite solutions and inequality constrained solutions to matrix equation AX=B, Hermitian equation AXA*=B or Hermitian equations; based on these, then, we come to investigate the numerical iterative algorithms for matrix inequalities; finally, we consider the iterative algorithms for the Re-nnd solutions to equations AX=C and AXB=C respectively, and the iterative algorithm for the common Re-nnd solution to AX=C and XB=D. By this project, we expect to develop some useful numerical iterative methods for the inequality constrained solutions of matrix equations and matrix inequalities, and advance the research on matrix theories and applications.

矩阵方程的半正定解和矩阵不等式分别在电力系统、多元线性回归模型、控制论等领域有着非常重要的应用，而针对这些矩阵方程和不等式的解的数值迭代算法的研究还较少。本项目将改进经典的矩阵迭代算法，拟建立矩阵方程AX=B、Hermitian型方程AXA*=B或Hermitian型方程组的半正定解和不等式约束解的数值迭代算法；在此基础上，继续探讨Hermitian型矩阵不等式的数值解的迭代算法；最后，研究矩阵方程AX=C和AXB=C的Re-nnd解的有效迭代算法、以及方程组AX=C和XB=D的公共Re-nnd解的迭代算法。期望通过本项目的研究，为矩阵方程的不等式约束解、矩阵不等式的数值研究提供可借鉴的方法，促进矩阵理论与应用研究的发展。

首先，项目组主要就秩约束下的矩阵最佳逼近问题展开了研究，利用奇异值分解与谱分解的理论，以及保范扩张原理，得到了几类逼近问题的Hermitian解与半正定解；其次，研究了可逆张量与其逆张量、相似张量的特征值与特征向量的关系，以及得到了几个更为有效的关于压电张量的谱半径的估计；最后，对于矩阵方程与矩阵不等式的求解做了一些研究，研究了矩阵方程AX=B的Hermitian自反解与Hermitian反自反解、以及半正定自反解，Yang-Baxter-like矩阵方程幂等解，以及矩阵不等式AXB+(AXB)*≥C的解。