可逆矩阵的高次伴随矩阵

用数学归纳法推出了可逆矩阵的高次伴随矩阵的公式,并结合可逆矩阵的基本公式得出了可逆矩阵的高次伴随矩阵的行列式和逆矩阵,给出了可逆矩阵的高次伴随矩阵的特征值和特征向量的表示公式,最后讨论了若干个可逆矩阵的乘积的高次伴随矩阵.     

二阶分块矩阵的伴随矩阵

本文主要讨论二阶分块矩阵的伴随矩阵,考虑到任何矩阵无论是否可逆,均存在伴随矩阵,将文献[1]中可逆的情况推广到了较一般情况,得到了二阶分块矩阵伴随矩阵的有关结论,并改进了文献[2]中相关结论的证明过程.     

矩阵多项式的平方根矩阵

研究了矩阵多项式的开平方问题,给出了矩阵多项式能开平方的充分必要条件及其平方根矩阵的个数,包含并推广了文[1]中的主要结论.     

Pascal矩阵与Vandermonde矩阵的关系

EI-Mikkawy M证明了对称Pascal矩阵Q_n和Vlandermonde矩阵V_n之间满足矩阵方程Q_n=T_nV_n,这里T_n是一个随机矩阵。本文证明了随机矩阵T_n能够分解成第一类Stirling矩阵和对角矩阵的乘积,得到了矩阵T_n的元素之间的递推关系,从而回答了EI-Mikkawy M的一个公开问题。同时得到了一些与Stirling数相关的组合恒等式。     

基于排列矩阵的一类析取矩阵

定义了k阶排列矩阵和(r+d)阶r-排列矩阵的概念,利用k阶排列矩阵和r-排列矩阵研究了d—析取矩阵、(d,e)-析取矩阵、(d,r,z]-析取矩阵的构造及其行数的行界.     

关于伴随矩阵的Jordan形矩阵

给出从一个矩阵的Jordan形矩阵和最小多项式求解它的伴随矩阵的Jordan形矩阵和最小多项式的方法.     

格矩阵的行列式与伴随矩阵

研究了格矩阵的行列式与伴随矩阵,给出了它们的一些代数性质,同时给出了由一个格矩阵构造一个传递矩阵的方法.     

用试验数据修正质量矩阵,阻尼矩阵与刚度矩阵

讨论用试验数据修正振动系统的双对称质量矩阵,阻尼矩阵与刚度矩阵的问题.依据特征方程,质量矩阵,阻尼矩阵与刚度矩阵的双对称性,利用代数二次特征值反问题的理论和方法,研究了这个问题解的存在性与惟一性,提出了修正质量矩阵,阻尼矩阵与刚度矩阵的一个新方法.利用矩阵的奇异值分解和矩阵的Kronecker乘积研究了方程的双对称解.给出了二次特征值反问题双对称解的一般表达式,讨论了对任意给定矩阵的最佳逼近问题,并给出了问题的最佳逼近解.     

循环矩阵与矩阵对角化

循环矩阵与矩阵对角化赵继安（甘肃省碌曲县中学７４７２００）矩阵的相似关系具有反身性，对称性和传递性，所以相似关系也是一种等价关系．按相似关系将复数域Ｃ上的ｎ阶矩阵分类，凡属同一个类的矩阵彼此相似，属于异类的任意两个矩阵都不相似，这样的类称为相似类．本...     

Nonderotary矩阵

Nonderotary矩阵是一类重要的矩阵,将从相似标准形、中心化子、相似类维数等角度刻划这类矩阵的性质,证明矩阵A是Nonderotary当且仅当与A可交换的所有矩阵都可以写成A的多项式,当且仅当A的相似类的维数最大.     

关于一类分块矩阵弱相似性质的理论探讨

本文提出了分块矩阵弱相似的概念,探讨了分块矩阵弱相似的条件和性质,得到了若干有意义的结果,为解决有关实际问题提供了一定的理论依据。     

B矩阵对策

提出了一类局中人都设定有得失控制值的二人零和有限对策,即B矩阵对策.引入了稳妥策略、弱稳妥策略等概念.给出了B矩阵对策的数学模型、有关理论和求解方法.最后,还给了一个计算例子.     

相应于Borcherds-Cartan矩阵的双参数弱Hopf代数

In this paper a class of two-parameter weak Hopf algebras wrτ,s（g） corresponding to Borcherds-Cartan matrix is constructed. This class consists of noncommutative and noncocommutative weak Hopf algebras but not Hopf algebras. It can be viewed as a generalized class of one-parameter weak Hopf algebras wUq（g）.     

关于不可约弱广义对角优势阵

本讨论了不可约弱广义对角优势阵的非奇异性及其特征值的分布,进而给出了非奇异Ｍ－阵的几个充分条件。     

The Galois Theory of Matrix C-rings

A theory of monoids in the category of bicomodules of a coalgebra C or C-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a C-ring (termed a matrix C -ring) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite injector. Based on this, the notion of a Galois module is introduced and the structure theorem, generalising Schneider’s Theorem II [Schneider, Isr. J. Math., 72:167–195, 1990], is proven. This is then applied to the C-ring associated to a weak entwining structure and a structure theorem for a weak A-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or infinite matrix ring contexts) is outlined. A Galois connection associated to a matrix C-ring is constructed. Dedicated to Stef Caenepeel on the occasion of his 50th birthday.     

矩阵最小二乘问题的迭代求解及其在机器翻译中的应用

研究一类线性矩阵方程最小二乘问题的迭代法求解,利用目标函数与矩阵迹之间的关系构造了矩阵形式的"梯度"下降法迭代格式,推广了向量形式的经典"梯度"下降法,并引入了两个矩阵之间的弱正交性来刻画迭代修正量的特点.作为本文算法的应用,给出了机器翻译优化问题的一种迭代求解格式.     

Bounds for the Singular Values of a Matrix

A lower-bound theorem is developed for the singular values ofa matrix A (and therefore for the eigenvalues of B = A^HA). Itis found that there is always a unique column scaling of A whichproduces the optimum bound. However, sharper bounds still maysometimes be obtained by taking advantage of matrix partitioning.It is shown that the resulting bounds may often (but not always)be better than those obtained by applying Gerschgorin's theoremto B. The equivalent upper-bound theorem is found to be weak.     

Matrix shrinkage of high-dimensional expectation vectors

The shrinkage effect is studied in estimating the expectation vector by weighting of mean vector components in the system of coordinates in which sample covariance matrix is diagonal. The Kolmogorov asymptotic approach is applied, when sample size increases together with the dimension, so that their ratio tends to a constant. Under some weak assumptions on the dependence of variables, the limit expression for the principal part of the quadratic risk function is found in dependence of weighting function. It is proved that the limit risk function does not depend on distributions. The extremum problem is solved, and an approximately unimprovable distribution-free estimator of the expectation vector is proposed.     

一四元数矩阵方程组的最佳逼近解

利用四元数矩阵的广义Frobenius范数和弱圈积,建立一个关于四元数矩阵的实函数并简洁表征其极小值.再用四元数矩阵的奇异值分解和广义Frobenius范数的性质,讨论四元数矩阵方程组[AX,XB]=[C,D]的最小二乘解,得到了解的具体表达式.最后在该方程组的解集合中导出了与给定矩阵的最佳逼近解的表达式.     

Matrix Models from Operators and Topological Strings

We propose a new family of matrix models whose 1/N expansion captures the all-genus topological string on toric Calabi–Yau threefolds. These matrix models are constructed from the trace class operators appearing in the quantization of the corresponding mirror curves. The fact that they provide a non-perturbative realization of the (standard) topological string follows from a recent conjecture connecting the spectral properties of these operators, to the enumerative invariants of the underlying Calabi–Yau threefolds. We study in detail the resulting matrix models for some simple geometries, like local \({\mathbb{P}^2}\) and local \({\mathbb{F}_2}\), and we verify that their weak ’t Hooft coupling expansion reproduces the topological string free energies near the conifold singularity. These matrix models are formally similar to those appearing in the Fermi-gas formulation of Chern–Simons matter theories, and their 1/N expansion receives non-perturbative corrections determined by the Nekrasov–Shatashvili limit of the refined topological string.