分块半正定矩阵若干新的特征值与奇异值关系研究

矩阵的特征值与奇异值是矩阵分析的主要研究对象，也是数值线性代数研究的核心问题之一。基于近年来人们对华罗庚矩阵以及更一般的positive partial transpose (PPT)矩阵的特征值与奇异值的研究，本项目将进行以下两方面的工作：一是建立几类新的关于2x2块半正定矩阵各个块矩阵间的特征值与奇异值的关系，这些关系涉及到主对角块矩阵的算术平均与几何平均。首先我们给出这几个新的关系式之间的内在联系，然后我们推导满足这些新的关系式的矩阵表示。二是给出华罗庚行列式不等式的其他等价形式，这些等价形式以压缩矩阵的特征值或奇异值的形式出现，进而给出这些等价形式的各种改进或加强。PPT矩阵在量子信息计算的相关研究起了很重要的作用，但是对它的特征值的数学性质及其刻画人们还研究的不够深入。本项目的研究将丰富矩阵分析理论并且为量子信息计算提供新的工具和视角。

The eigenvalues and singular values of matrices are the main objects of study in matrix analysis; they also form one of the central problems in numerical linear algebra. Based on the recent study on the eigenvalues and singular values of the Hua matrix or more generally the positive partial transpose (i.e., PPT) matrices, in this project we would propose researches on the following two aspects: the first one is to establish some new relations between eigenvalues and singular values associated with 2x2 block positive semidefinite matrices. These new relations involve the arithmetic mean and geometric mean of the main diagonal blocks. First of all, we investigate the intrinsic connections between these new relations, and then we derive the matrix representations for which these new relations are met. The second one is to present other equivalent forms of the Hua determinantal inequality, these equivalent expressions are in terms of the eigenvalues or singular values of contractive matrices; and then we present the improvement or strengthening of these equivalent forms. PPT matrices play an important role in the field of quantum information and computing, but the mathematical properties or characterizations of their eigenvalues are not well studied. The implement of this present project will stimulate further theoretical development of matrix analysis and will find potential applications or provide new perspectives in the area of quantum information and computing.