半定内积空间下矩阵分解的理论及应用研究

This projector will study the theory of matrix factorizations under the semidefinite inner product space and its applications in the quadratic programming and statistics. Firstly, we consider the properties of self-adjoint matrices under the semidefinite inner product space which are similar to Hermitian matrices under the classical inner product space for improvements in the existing singular value decomposition of matrices under the semidefinite inner product space. Secondly, we will propose polar factorization and QR factorization of matrices under the semidefinite inner product space according to the experience that the classical factorizations of matrices were generalized to the weighted factorizations of matrices and the indefinite factorizations of matrices and the properties of matrices under the semi-definite inner product space, and then discuss their properties, existence and uniqueness conditions and calculation methods. In addition, we will consider the absolute perturbation bounds, relative perturbation bounds, multiplicative perturbation bounds, the first perturbation bounds and rigorous perturbation bounds of the new proposed polar factorization and QR factorization of matrices under the semidefinite inner product space using the classical matrix equation approach, the refined matrix equation approach, the matrix vector equation approach and their combinations, respectively. Finally, we will discuss the applications of the new proposed singular decomposition, polar decomposition and QR factorization of matrices under the semidefinite inner product space in the quadratic programming problem and the problem in statistics.

本项目拟研究半定内积空间下矩阵分解的理论及其在二次规划与统计学上的应用。首先，研究半定内积空间下自伴随矩阵的类似于经典内积下Hermite矩阵的性质，从而完善现有的半定内积下的矩阵奇异值分解；其次，借鉴经典内积下的矩阵分解推广到正定内积与不定内积下矩阵分解的经验，同时结合半定内积下矩阵的性质来构建半定内积空间下的矩阵极分解与QR分解，并探讨它们的性质、存在性与唯一性条件及计算方法；再次，利用经典的矩阵方程的方法、精致的矩阵方程的方法、矩阵向量方程的方法及它们的组合分别研究新构建的半定内积空间下矩阵极分解与QR分解的绝对扰动界、相对扰动界、乘法扰动界与一阶扰动界、严格扰动界；最后，讨论新构建的半定内积空间下的矩阵奇异值分解、极分解与QR分解在二次规划与统计学上的应用。

半正定内积空间下的矩阵分解是经典内积空间中矩阵分解的拓展，不但本身具有重要的理论研究价值，而且在优化和统计学上具有重要应用。.本项目主要研究了在半正定内积空间下矩阵分解的理论分析，优化及半正定矩阵的不等式等问题作了一些相关研究。具体研究内容如下：.（1）提出半正定内积空间下的一些矩阵极分解并研究了它们的存在唯一性条件及计算方法等，相关结果发表在SCI期刊《Linear and Multilinear Algebra》；.（2）建立非凸强向量变分不等式的误差界及研究了非凸强向量均衡问题解集的连通性与路径连通性等问题，相关结果发表在SCI期刊《Optimization》和《Journal of Optimization Theory and Applications》；.（3） 推广了一些经典矩阵不等式，比如：Fischer不等式， 相应结果发表在SCI期刊《Linear Algebra and its Applications》和《Mathematical Inequalities & Applications》上；.（4）通过研究分块半正定矩阵与分块正定偏转置(PPT)矩阵之间的关系，补充了一些半正定矩阵之间的不等式，相应结果发表在SCI期刊《Linear Algebra and its Applications》；.（5）通过研究矩阵特征值的初等对称函数的相关性质，给出了一个半正定分块矩阵的行列式不等式，该不等式统一了1961年Thompson的一个结果和1994年Fiedler和Markham的一个结果；相应结果发表在SCI期刊《Linear Algebra and its Applications》。