四元数体上Hermitian矩阵空间保秩1的加法映射

Let Q be the quaternion division algebra over real field F.Denote by H_n(Q)the set of all n×n hermitian matrices over Q.We characterize the additive maps from H_n(Q) into H_m(Q)that preserve rank-1 matrices when the rank of the image of I_n is equal to n. Let Q_R be the quaternion division algebra over the field of real number R.The additive maps from H_n(Q_R) into H_m(Q_R)that preserve rank-1 matrices are also given.     

自共轭四元数矩阵特征值的极值原理

本文将Ｈｅｒｍｉｔｅ算子特征值的两个一般的极值原理推广到自共轭四元数矩阵，并给出了几个推论，同时也去掉了子空间的包含条件     

关于广义四元数代数上矩阵的迹的一个等式

众所周知,相似矩阵的迹相等对于非交换代数和环上的矩阵不一定成立,有趣的问题是给定一个条件使得相似矩阵的迹相等对于非交换代数或非交换环上的矩阵成立。本文对于特征不是2的任意域F上定义的广义四元数代数上的两个矩阵A和B,给出如果A和B相似并且它们的主对角线上的元素在F中,那么它们的迹相等。     

Some New Bounds for Singular Values and Eigenvalues of Matrix Products

For two Hermitian matrices A and B, at least one of which is positive semidefinite, we give upper and lower bounds for each eigenvalue of AB in terms of the eigenvalues of A and B. For two complex matrices A,B with known singular values, upper and lower bounds are deduced for each singular value of AB.     

可除的四元数代数上多项式环的Grobner基

设F是一个特征不等于2的域,A是,上的一个可除代数。本文研究了A上多项式环A[x1,X2,…,xn]中理想是有限生成的,以及它的Grobner基;也表明F[x1,x2,…,xn]中有限子集G是F[x1,x2,…,xn]的Griobner基当且仅当G是A[x1,x2,…,xn]中的Grobner基。     

关于四元数EP矩阵偏序的研究

本文利用四元数矩阵的奇异值分解给出四元数EP矩阵的一个刻画,并得到四元数EP矩阵减序,左（右）星序,星序的相应刻画定理与性质定理．     

Bounds for Determinants of Quaternion Matrices

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自共轭四元数循环矩阵的特征值问题

循环矩阵是一类应用广泛的特殊矩阵.设A是一个自共轭四元数循环矩阵,运用四元数矩阵的复表示,以及循环矩阵的特定结构形式,得到了矩阵A的特征值的计算公式.反之,对于任意给定的n个实数,证明了一定存在自共轭四元数循环矩阵A,使得A以这n个实数为它的特征值,同时给出了自共轭四元数循环矩阵A的计算方法.推广了复循环矩阵的相关理论结果.     

On Eigenvalues of Split Quaternion Matrices

The main purpose of this paper is to set a method of finding eigenvalues of split quaternion matrices. In particular, we will give an extension of Gershgorin theorem, which is one of the fundamental theorems of complex matrix theory, for split quaternion matrices.     

四元数自共轭矩阵乘积的特征值不等式

由于四元数对乘法无交换律,因而对四元数自共轭矩阵的特征值问题的讨论比复数矩阵的相应问题要困难得多,文[1]、[2]分别对四元数自共轭矩阵的特征值和两个四元数自共轭矩阵乘积的特征进行了估计,做了一定的工作,但与复数域上的有关结果相比较,还有较大差距.本文对四元数自共轭矩阵乘积的特征值进行了探讨.得到了较好的结论,推广了[1]、[2]中的结果。     

四元数自共轭矩阵和的特征值的不等式

本文将Wietandt关于复自共轭矩阵的特征值和的变分特征以及和的特征值的不等式推广到四元数体上，由此还给出了复自共轭矩阵的主对角元与特征值的优化关系的Schur定理、复矩阵的主对角元的模与奇异值的优化关系的Fan定理在四元数体上的推广.     

Zeros of Two-Sided Quadratic Quaternion Polynomials

The structure and computation are given for the zeros of a two-sided quadratic quaternion polynomial. In particular, the computation formulae and the sharp bound of essential number are established for the zeros of a two-sided quaternion polynomial of the form \({u^2 + puq + r (p, q, r \in \mathbb{H})}\) . As applications, some known results are improved and a conjecture presented by Janovská and Opfer is answered.     

Upper Bounds for the Laplacian Graph Eigenvalues

We first apply non-negative matrix theory to the matrix K = D A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1 (G) of G and the spectral radius p(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extremal graphs which achieve the upper bounds.     

The Splitting Representation of Matrix and Matrix Equation over a Central Division Algebra

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Bounds for the Eigenvalues of Non-selfadjoint Differential Operators

A generalization of the Gerschgorin circle theorem is appliedto an infinite matrix representation of a differential operatorto obtain bounds for its eigenvalues.     

Hermite矩阵特征值的新扰动界

本文研究了Hermite矩阵特征值的任意扰动,给出了新的绝对和相对扰动界．所给出的界改进了Hoffman-Wielandt和Kahan早期的结果．     

双调和算子本征值的上界

设Ω是 Rn中的有界区域 ,其边界足够光滑 ,λk为双调和算子在自由边界条件下的第 k个本征值 ,利用变分原理及 Fourier变换 ,给出了本征值部分和 ∑kj=1λj的一个上界 ,该上界仅依赖于区域的体积 .     

Extrinsic Bounds for Eigenvalues of the Dirac Operator

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.