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．     

正定自共轭矩阵行列式的一个极小值和Hadamard不等式的再改进

本文首先按Ｄｉｅｕｄｏｎｎｅ意义下行列式给出了正定自共轭矩阵行列式的一个极小值。进而，改进了Ｈａｄａｍａｒｄ不等式，并指出，按谢邦杰意义下行列式，有类似结论，推广了［１］－［４］的结果。     

一道行列式证明题的多种证明方法

针对线性代数教材中一道行列式证明题，利用行列式的性质，给出多种证明方法，旨在启发学生对相关行列式计算或证明题的解题方法进行探索．     

四元数矩阵的实值行列式

本文定义了任一个四元数方阵的实值行列式．实值行列式有一些好的性质，它是按照谢邦杰定义的可中心化四元数矩阵的行列式的推广．     

行列式理论的一种建立方法

引进行列式的一种公理化定义，并由此建立行列式理论，弥补了教材[1]中相关内容存在的缺陷。     

三对角行列式及其应用

利用递归方程得到了计算三对角行列式的一般方法，研究了三对角行列式在线性代数及组合数学中的应用。     

复亚正定矩阵的几个行列式不等式

讨论了复亚正定矩阵的几个行列式不等式的问题，并在复亚正定矩阵的基础上推广了几个行列式不等式，修正并推广了现有的结果。     

关于一类对称方程组非零解的判定

张远达教授在[1]中利用范得蒙行列式判明了对称方程组只有零解，本言语要利用所谓增次的范得行列式     

“杨辉三角”中某些矩阵及其行列式的讨论

“杨辉三角”中某些矩阵及其行列式的讨论张之正，刘麦学（河南洛阳师范专科学校数学系４７１０２２）近来，许多作者对杨辉三角中的矩阵和行列式进行了讨论（如［１］─［５］），文［６］还给出了一类矩阵的逆矩阵的求法．本短文对这些矩阵及行列式做进一步的探讨．杨辉...     

四元数阵重行列式与复阵行列式的关系

指出四元数阵重行列式可用复阵行列式来表示，于是，复阵的伴随矩阵、求逆阵公式、秩的下界等，都可相应地推广到四元数阵．     

复正定矩阵的Minkowski不等式

建立了复正定矩阵的几个行列式不等式,将正定Ｈｅｒｍｉｔｅ阵的Ｍｉｎｋｏｗｓｋｉ不等式、 Ｏｓｔｒｏｗｓｋｉ－Ｔａｕｓｓｋｙ不等式推广到了复正定矩阵上,推广改进了一些文献的结果．     

REFINEMENTS OF THE FAN-TODD'S INEQUALITIES

Refinements to inequalities on inner product spaces are presented. In this respect, inequalities dealt with in this paper are: Cauchy's inequality, Bessel's inequality, Fan-Todd's inequality and Fan-Todd's determinantal inequality. In each case, a strictly increasing function is put forward, which lies between the smaller and the larger quantities of each inequality. As a result, an improved condition for equality of the Fan-Todd's determinantal inequality is deduced.     

REFINEMENTS OF THE FAN－TODD‘S INEQUALITIES

51. IntroductionIn recent years, refinements or interpolations have played an important role on severaltypes of inequalities with new results deduced as a consequence. Please refer to the papers[2, 8, 9, 12], etc. The aim of this paper is to furnish refinements of the Cauchy's and Bessel'sinequalties as shown in Section 2, and also refinements of the Fan-Todd's inequality and theFan-Todd's determinantal inequality in Sections 3 and 4, with an improved condition forequality derived.First of…     

On a generalized Fan inequality

It is shown that the ω- and τ-matrices, the weakly sign symmetric matrices, the R- and V-matrices, and the matrices c-equivalent to an M-matrix or to a real matrix with nonpositive off-diagonal elements, can all be characterized by the same determinantal inequality, which we call a generalized Fan inequality.     

A Reverse of bessel’s inequality in 2-inner product spaces and some grüss type related results with applications

A reverse of Bessel’s inequality in 2-inner product spaces and companions of Grüss inequality with applications for determinantal integral inequalities are given.     

关于《余A-G型Ky Fan不等式》的发展

Two new proofs of a discrete Ky Fan inequality of the comple mentary A-G type are given. Its continuous version and determinantal analogue on a set of pairwise commutative positive definite matrices are established. A further extension concerning general positive definite matrices of the inequality is also suggested.     

An extension of Harnack type determinantal inequality

We revisit and comment on the Harnack type determinantal inequality for contractive matrices obtained by Tung in the nineteen sixties and give an extension of the inequality involving multiple positive semidefinite matrices .     

Erratum: the robertson-taussky inequality revisited

A counterexample is given to the permanental analog (of the titled determinantal inequality) asserted in the titled paper. The fault lay with an invalid identity used in its proof.     

Polynomials with and without determinantal representations

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov [9] have proved that any real zero polynomial in two variables has a determinantal representation. Brändén [2] has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence.     

Fischer determinantal inequalities and Highamʼs Conjecture

We obtain a Fischer type determinantal inequality for matrices with given angular numerical range. We discuss the growth factor for Gaussian elimination for linear systems in which the coefficient matrix has this form and give a proof of Higham?s Conjecture.