一类Hermitian型矩阵不等式方程的研究及其应用

This project mainly study solving the matrix inequality equations related to Löwner partial order, and consider its applications in common Re-nnd solutions, common Re-pd solutions, Re-nnd generalized inverses and constrained matrix equations. Firstly, we study the existence and expressions of the solution to matrix inequality AXB+(AXB)^* ≥C, where C is Hermitian, for applications, we also investigate the common Re-nnd solutions, common Re-pd solutions to AX=C and XB=D, Re-nnd generalized inverses. Thirdly, we consider the solutions to matrix equality equations with some inequality constraints such as EXE^* ≥(≤)D, EXF+(EXF)^* ≥(≤)D. Finally, for inequality equations such as AXA^* ≥(≤)B and CXC^* ≥(≤)D, we will discuss the existence of common Hermitian solutions and common nonnegative-definite solution, including the expressions of these solutions. Also consider its applications in statistics, control system, optimization theory.

本项目主要研究Löwner偏序下的矩阵不等式方程的求解，及其在方程组的公共Re-nnd解、Re-pd解、方阵的Re-nnd广义逆、约束方程等方面的应用。首先研究不等式方程AXB+(AXB)^* ≥C解的存在性以及解的一般表达式，这里C为Hermitian矩阵，同时将此不等式的解应用到求解方程组AX=C和XB=D的公共Re-nnd解和Re-pd解的一般表达式，以及构建方阵的Re-nnd广义逆。其次，研究矩阵等式方程在形如EXE^* ≥(≤)D、EXF+(EXF)^* ≥(≤)D不等式约束条件下的约束解。最后，针对矩阵不等式方程组解的存在性以及解的结构展开研究，例如探索形如AXA^* ≥(≤)B和CXC^* ≥(≤)D等方程组存在公共Hermitian解和公共半正定解的充分必要条件，以及这两种公共解的一般表达式。同时探讨其在统计、控制系统、优化理论等领域的应用。

本项目主要研究了Löwner偏序下的矩阵不等式的求解，及其在方程的Re-nnd解、方阵的Re-nnd广义逆、约束方程等方面的应用。首先研究不等式AXB+(AXB)^* ≥C解的存在性以及解的一般表达式，这里C为Hermitian矩阵，(A B^*)为行满秩矩阵，同时将此不等式的结论应用到求解方程AXB=C的Re-nnd解的一般表达式，以及构建方阵的Re-nnd {1,2,3}-逆和 {1,2,4}-逆。其次，研究了方程AXA^* =B或AX=B在不等式约束CXC^* ≥(≤)D下的约束解。最后，也研究了矩阵方程AX=B的自反解与反自反解，方程组AX=B和XC=D的公共自反解与反自反解，以及相应的矩阵最佳逼近问题。