关于正定矩阵的子式阵的正定性

研究正定矩阵的子矩阵,利用合同标准形分别给出了复正定矩阵的子式阵为复正定矩阵和实正定矩阵的子式阵为实正定矩阵的充分必要条件,其结果简单而实用.     

关于合成矩阵的正定性

利用标准形分别给出了复正定矩阵的合成矩阵为复正定矩阵和实正定矩阵的合成矩阵为实正定矩阵的充分必要条件,其结果简单而实用.     

正定矩阵的子式阵仍是正定矩阵的一个充要条件

文[2]证明了实对称正定矩阵的子式阵仍然是实对称正定矩阵,文[3]给出了一般的正定矩阵的的概念,本文利用标准型给出了一般正定矩阵的子式阵仍然是正定矩阵的充要条件.     

正定矩阵的子式阵仍是正定矩阵的一个充要条件

文[2]证明了实对称正定矩阵的子式阵仍然是实对称正定矩阵,文[3]给出了一般的正定矩阵的的概念,本文利用标准型给出了一般正定矩阵的子式阵仍然是正定矩阵的充要条件．     

实正定矩阵的复合矩阵的正定性

本文讨论了实正定矩阵的复合矩阵的正定性，并且给出了实正定矩阵的复合矩阵仍为正定矩阵的一个充要条件.     

复正定矩阵的充要条件

本文研究Hermite正定矩阵、实正定矩阵的推广概念复正定矩阵,给出了复方阵复正定性的一些充分必要条件。     

次正定复矩阵次Schur补的一些性质

利用复矩阵的Schur补和次正定性,研究了次正定复矩阵的次Schur补的一些性质,得到了次正定复矩阵次Schur补的几个行列式不等式,将相关文献的相应结果由次正定次Hermite矩阵推广到次正定复矩阵.     

一类广义正定矩阵的性质

广义正定矩阵有一系列推广,研究了一类广义正定矩阵,得到了这类广义正定矩阵的一些性质,并利用这些性质得到了这类广义正定矩阵的一些行列式不等式.     

关于复矩阵乘积的正定性

两复正定矩阵之和必是复正定矩阵,但其积未必是复正定矩阵.研究了复矩阵之积的正定性,给出了复矩阵之积为复正定矩阵的一系列判定条件,获得了一些新的结果,改进并推广了K y Fan T aussky定理及Fe jer定理.     

线性互补问题中几类特殊矩阵与半正定矩阵之间的关系

线性互补问题中特殊矩阵M 的性质是线性互补问题中研究的重要部分之一,本文深入研究了Cf0矩阵与半正定矩阵、子正定矩阵与半正定矩阵之间的关系,并且得到了特殊矩阵是半正定矩阵的一些充分条件。     

Almost strictly totally positive matrices

A determinantal identity, frequently used in the study of totally positive matrices, is extended, and then used to re-prove the well-known univariate knot insertion formula for B-splines. Also we introduce a class of matrices, intermediate between totally positive and strictly totally positive matrices. The determinantal identity is used to show any minor of such matrices is positive if and only if its diagonal entries are positive. Among others, this class of matrices includes B-splines collocation matrices and Hurwitz matrices.This author acknowledges a sabbatical stay at IBM T.J. Watson Research Center in 1990, which was supported by a DGICYT grant from Spain.     

泛正定阵与广义正定阵的关系

本文利用矩阵的稳定值,给出泛正定阵的等价条件,进一步得到泛正定阵与广义正定阵之间的一些关系．     

实正定矩阵与Minkowski不等式的再推广

本文给出实广义正定矩阵概念的新推广及其基本性质,讨论它及常见几种定义下广义正定矩阵的代数结构,得到非对称正定矩阵乘积的一个新刻画,并利用所获广义正定矩阵的性质,拓广了Minkowski,OstrowskiTaussky等矩阵不等式的取值范围.     

The realization of positive definite matrices via planar networks and mixing-type sub-cluster algebras

As an improvement of the combinatorial realization of totally positive matrices via the essential positive weightings of certain planar network by S.Fomin and A.Zelevinsky [7], in this paper,we give a test method of positive definite matrices via the planar networks and the so-called mixing-type sub-cluster algebras respectively,introduced here originally.This work firstly gives a combinatorial realization of all matrices through planar network,and then sets up a test method for positive definite matrices by LDU-decompositions and the horizontal weightings of all lines in their planar networks.On the other hand,mainly the relationship is built between positive definite matrices and mixing-type sub-cluster algebras.     

New results on the cp-rank and related properties of co(mpletely )positive matrices

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order and a further improvement for such matrices of order six.     

A new look at totally positive matrices

A close relationship between the class of totally positive matrices and anti-Monge matrices is used for suggesting a new direction for investigating totally positive matrices. Some questions are posed and a partial answer in the case of Vandermonde-like matrices is given.     

非负定阵元素的扰动和正定完备化

在保持非负定性不变的前提下,本文对矩阵每一元素容许多大的扰动作了进一步的研究, 将本文的结论和C．R．Johnson提出的部分正定阵的正定完备化进行比较,容易发现对已知的正定矩阵求扰动,本文的结论比用C．R.Johnson的正定完备化计算扰动形式上更简单,同时也给出了不同于C．R．Johnson的部分正定阵的正定完备化表示的另外一个公式,推出了这些正定完备化矩阵应具有的若干性质．     

The approximation of a totally positive band matrix by a strictly banded totally positive one

Every nonsingular totally positive m-banded matrix is shown to be the product of m totally positive one-banded matrices and, therefore, the limit of strictly m-banded totally positive matrices. This result is then extended to (bi)infinite m-banded totally positive matrices with linearly independent rows and columns. In the process, such matrices are shown to possess at least one diagonal whose principal sections are all nonzero. As a consequence, such matrices are seen to be approximable by strictly m-banded totally positive ones.     

The Koteljanskii inequalities for mixed matrices

Recently, a new class of matrices, called mixed matrices, that unifies the Z-matrices and symmetric matrices has been identified. They share the property that when the leading principal minors are positive, all principal minors are positive. It is natural to ask what other properties of M-matrices and positive definite matrices are enjoyed by mixed matrices as well. Here, we show that mixed P-matrices satisfy a broad family of determinantal inequalities, the Koteljanskii inequalities, previously known for those two classes. In the process, other properties of mixed matrices are developed, and consequences of the Koteljanskii inequalities are given.