两类四元数矩阵偶的GH合同标准形

该文给出两类四元数矩阵偶〈A，B１〉与〈A，B２〉的ＧＨ 合同标准形，其中A为半正定自共轭阵，B１ 为斜自共轭阵，B２ 为自共轭阵．由此分别得到（广义）半正定与正定四元数矩阵的ＧＨ合同标准形，以及矩阵同时对角化问题的若干个结果．     

四元数体上重行列式的性质及其应用

本文得到了四元数体上重行列式的一些基本不等式，给出了矩阵为正定自共轭阵时，行列式与重行列式的显式关系，同时也给出了文［１］中行列式与文［５］中行列式两种定义的关系。提出了四元数体上广义正定自共轭阵的概念，并获得了这类阵的基本性质．     

两个四元数自共轭矩阵乘积的特征值估计

设A和B都是四元数自共轭半正定矩阵，或者其中之一是正定的，而另一个是自共轭的，本文改进并推广了[2]对乘积AB的特征值估计.     

四元数自共轭矩阵乘积的相似变换

本文证明了下列结果:(i)四元数矩阵A可写成两个自共轭四元数矩阵的乘积A相似于实矩阵A Hermite相似于A~*.(ii)A可写成一个半正定自共轭四元数矩阵与一个自共轭四元数矩阵的乘积A相似于实对角矩阵或者A～diag(D,I_r(×)J_2(O)),其中D是一个实对角矩阵.本文还给出了体上实矩阵AB与BA相似的一个充要条件.     

四元数体上共轭辛矩阵的结构及应用

把实数域上的辛矩阵概念推广到四元数体上形成共轭辛矩阵类.用矩阵四分块形式刻划了正定辛矩阵和自共轭辛矩阵的特征结构.作为应用,给出四元数矩阵方程AS=B存在四分块对角型共轭辛矩阵解的充要条件及其解的表达式,同时用数值算例说明所给方法的可行性.     

四元数体上的Marshall-Olkin不等式及改进

本文证明了四元数自共轭半正定矩阵乘积的一些不等式.这些结果推广、改进了复数域上的Marshall-Olkin不等式.     

正定自共轭矩阵行列式的含参数的上界

本文绘出几个关于正定自共轭矩阵行列式的含参数的上界,从而在实四元数除环上进一步推广了Hadamard定理。     

关于矩阵的Bergstrom型不等式的修正

本文指出了[1]中关于实数域上亚正定阵和[3]中关于四元数体上正定自共轭矩矩阵的Bergstrom型不等式推广的错误,同时给出有关结论的正确形式.     

四元数体上的矩阵方程

<正> 如所周知,矩阵方程是矩阵研究中的重要方向之一,但四元数体上的矩阵方程的研究至今仍然少见.注意到四元数矩阵研究的新近进展,本文对此作了一些研究,主要目的是阐述上矩阵方程AX=B的正定自共轭解、矩阵方程AX+XA=B与     

关于四元数矩阵的行列式不等式

本文证明了正定自共轭四元数矩阵的行列式的一些高精度的不等式,并得到著名的 Hadamard 不等式新的改进形式,同时也改进了谢邦杰等人的结果.     

Norm inequalities for matrix geometric means of positive definite matrices

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.     

Principal powers of matrices with positive definite real part

We study principal powers of complex square matrices with positive definite real part, or with numerical range contained in a sector. We extend the notion of geometric mean to such matrices and we establish an operator norm bound in this context.     

关于算术、对数、指数、几何平均不等式的概率证法

本文用概率方法证明了关于两个相异的正数的几何平均、对数平均、指数平均和算术平均的不等式关系 .     

A fast iterative method to find the matrix geometric mean of two HPD matrices

The purpose of this research is to present a novel scheme based on a quick iterative scheme for calculating the matrix geometric mean of two Hermitian positive definite (HPD) matrices. To do this, an iterative scheme with global convergence is constructed for the sign function using a novel three‐step root‐solver. It is proved that the new scheme is convergent and shown to have global convergence behavior for this target, when square matrices having no pure imaginary eigenvalues. Next, the constructed scheme is used and extended through a well‐known identity for the calculation of the matrix geometric mean of two HPD matrices. Ultimately, several experiments are collected to show its usefulness.     

无限区间上的微分中值定理

以微分中值定理的几何意义作为切入点,将微分中值定理的适用区间从原来的有限区间[a,b]推广到无限区间（-∞,＋∞）,并给出相应的推广结论．     

On a new construction of geometric mean of -operators

For n positive definite operators A₁,…,A_n, Ando-Li-Mathias defined geometric mean of n-operators by symmetric procedure. It has many nice properties, and is studied by many authors. But the process is so complicated to compute. In this paper, we shall attempt to make a new construction of geometric mean of n-operators which we can compute it easier than geometric mean by Ando–Li–Mathias.     

正定Hermite矩阵加权幂平均的行列式不等式

本文给出了m个正定Hermite矩阵加权幂平均的行列式的一个不等式，它是m个正数的加权益平均不等式的自然推广，也是正定Hermite矩阵行列式的凸性不等式的推广．     

The harmonic and geometric mean of vectors

New vector operations are defined that generalize the classical harmonic and geometric means of scalars. The definition of the harmonic mean involves an explicit vector formula, whereas the geometric mean of two vectors is defined by an iterative procedure. Two special cases are described in some detail, the case of 2-vectors which are equivalent to the complex numbers, and the case of 4-vectors which are a representation of the quaternions.     

关于“四元数自共轭矩阵与行列式的几个定理”的注记

本刊1985年第4期发表郝稚传的“四元数自共轭矩与行列式的几个定理”(称为〔1〕)的文章的主要工作分为两部分。〔1〕在英文摘要中写到:“(i)This essay has improvedthe conclusion of theorem 8 and theorem 9 in 〔2〕”即〔1〕之第一部分工作在于改进了〔2〕的定理8、9的结果,其根据就是〔1〕的定理1、2.〔1〕称定理1—若     

Norm inequalities related to the matrix geometric mean

Inequalities for norms of different versions of the geometric mean of two positive definite matrices are presented.