矩阵可对角化的一个充要条件

本文给出矩阵可对角化(即可与对角矩阵相似)的一个充要条件,并推广了文[1]中的一个结果。首先叙述如下: 引理设A,B都是n阶矩阵,则有秩(AB)≥秩A+秩B-n 证明可见[2],这里从略。定理设A是数域F上的一个n阶矩阵,     

William E.Roth’s第二定理的一个新证明

仅利用矩阵的基本理论建立了四个引理,并且在此基础上对W.E.Roth第二定理给出了新的证明．     

Leray-Schauder不动点定理的一个新证明

给出Leray-Schauder不动点定理的一个新证明.我们首先给出集值映射的焊接引理,利用集值映射的焊接引理和Kakutani不动点定理证明Leray-Schauder不动点定理,并证明Leray-Schauder不动点定理与Brouwer不动点定理等价.     

一个线性代数定理证明的新方法

实对称矩阵合同相似于对角矩阵这一结论是实二次型理论中的一个基本定理．本文给出了证明这一定理的新方法，该方法简洁、清晰、通俗易懂．     

关于商空间的一个注记

《拓扑学基础》[1]中的定理4.1.5,(1)是商空间理论中商映射与映射族的乘积间关系的一个重要命题,但该定理(1)的证明存在欠妥之处.本文利用饱和集与管状引理给出了该定理中(1)的一个正确的证明,并介绍了它的一个重要应用.     

《Laffey－Choi定理的一个证明》一文中的错误及纠正

《Ｌａｆｆｅｙ－Ｃｈｏｉ定理的一个证明》一文中的错误及纠正黄礼平（湘潭矿业学院基础科学部４１１２０１）文［１］中给出了Ｌａｆｆｅｙ—Ｃｈｏｉ定理的一个更初等、更简明的“证明”，遗憾的是其中关键的引理中的证明是错误的，现摘录如下：“引理设Ａ，Ｂ是ｎ阶复...     

Hamilton-Cayley定理的一种证法

<正> Hamiltion-Cayley定理是矩阵代数中一个基本定理,有重要的理论和应用价值。本文提出一种简明的证法,它仅利用矩阵的Jordan标准形和矩阵的乘法运算规则。引理设T_i是对角线上第i个元素为零的n阶上三角矩阵(i=1,2,…,n)测     

实对称矩阵A正定的充分条件的证明

具体给出一个实对称矩阵A以后,判定A正定的有效方法由下述定理给出。 定理 实二次型f(x_1.…x_n)=sum from i,j=1 to n (a_(ij)x_ix_j)=X’AX是正定的充要条件是矩阵A的顺序主子式全大于零。 必要性的证明在此就不再赘述。下面我们     

三角形旁切圆切点的一个性质

定理　设△ABC的旁切圆⊙Ia、⊙Ib、⊙Ic 分别切BC、CA、AB于点X、Y、Z .过YZ和BC的中点X1和D作一直线X1D ,及类似的直线Y1E和Z1F(如图 1) .则X1D、Y1E、Z1F三线共点且该点恰为△DEF的内心 .先给出下面的引理 .引理 1[1] 　分别过三角形三边中点的三条周界平分线交于一点 ,这一点称为第二等周中心 (证明略 ) .图 1　　　　　　图 2引理 2　若四边形的一组对边相等 ,则相等的这一组对边交角的平分线必平行于另一组对边中点的连线 .证明　如图 2 ,设四边形ABCD中 ,AD=BC ,E、F分别为AB、CD的中点 ,AD、BC的延长线交于点…     

许宝騄在多元分析方面的工作

<正> 从1938年到1945年,许所发表的论文处在多元分析数学理论发展的前沿.可以猜想,他受到了与当时也在伦敦大学学院的 R.A.Fisher 的影响.1945年后,他在哥伦比亚大学和北卡罗来纳大学讲授多元分析,在那里他培养学生从事这一领域的研究.如同一个有高度素养的数学家那样,许推进了矩阵论在统计理论中的作用,同时证明丁关于矩阵的一些新的定理.     

关于离散型随机变量次序统计量分布矩阵对称性猜想的探索

本文研究了离散型随机变量次序统计量的分布矩阵的对称性 ,获得了二个定理 .定理 1　服从等概率二点分布或等概率三点分布的离散型随机变量的次序统计量的分布矩阵是对称矩阵 .定理 2　取值有限且等概率的离散型随机变量的次序统计量的分布矩阵具有中心对称性 .     

Some results on numerical ranges and factorizations of matrix polynomials

Some new factorization theorems for monic matrix polynomials are obtained. These theorems are based on the numerical range having the number of connected components equal to the degree of the polynomial. For second degree polynomials, sufficient conditions are given for the numerical range to have two connected components.     

The symmetry principle in continuum mechanics

For problems of the mechanics of an anisotropic inhomogeneous continuum, theorems are given concerning the uninterrupted symmetrical and antisymmetrical analytical continuation of the solution through the plane part of the boundary surface of the medium. Theorems are given for two types of mechanics problem; in the first of these both symmetrical and antisymmetrical continuations of the solution are allowed, while in the second only symmetrical continuation of the solution is allowed. Problems of the first type include problems which are reduced to linear thermoelastic dynamic differential equations of motion of an inhomogeneous anisotropic medium possessing a plane of elastic symmetry, to linear thermoelastic dynamic differential equations of motion of an inhomogeneous Cosserat medium, to non-linear differential equations describing the static elastoplastic stress state of a plate, etc. The second type includes problems which are reduced to non-linear differential equations describing geometrically non-linear strains of shells, to Navier–Stokes equations, etc. These theorems extend the principle of mirror reflection (the Riemann–Schwartz principle of symmetry) to linear and non-linear equations of continuum mechanics. The uninterrupted continuation of the solutions is used to solve problems of the equilibrium state of bodies of complex shape.     

Antieigenvalue techniques in statistics

In a number of his recent papers Karl Gustafson has outlined the similarities between the Antieigenvalue Theory he founded and several finite dimensional matrix optimization theorems for positive matrices arising in statistics. In this paper, we will show how the techniques that the author and Karl Gustafson have used for computation of Antieigenvalues can also be applied to prove and generalize these matrix optimization theorems in statistics. We will primarily focus on two techniques which we have used in Antieigenvalue computations in recent years. These two techniques are a two nonzero component property for certain class of functionals, and converting the matrix optimization problems in statistics to a convex programing problem. Indeed, these two techniques allow us to generalize some of the matrix optimization problems arising in statistics to strongly accretive operators on finite or infinite dimensional Hilbert spaces.     

LMI优化问题的择一性定理与对偶

考虑目标函数是线性函数约束条件为线性矩阵不等式的LMI优化问题,讨论了LMI优化问题中的四个择一性定理.每种类型的择一性定理包含两个线性不等式和(或)等式系统,一个原始系统和一个对偶系统.弱择一性定理说明两系统中至多只有其一有解;基于凸集分离理论得到的强择一性定理说明两系统有且仅有其一有解.并在此基础上推导了LMI优化...     

Entropy and the Uncertainty Principle

We generalize, improve and unify theorems of Rumin, and Maassen–Uffink about classical entropies associated with quantum density matrices. These theorems refer to the classical entropies of the diagonals of a density matrix in two different bases. Thus, they provide a kind of uncertainty principle. Our inequalities are sharp because they are exact in the high-temperature or semi-classical limit.     

A note on quaternion matrices and split quaternion matrix pencils

In this paper, localization theorems for left and right eigenvalues of a quaternion matrix are presented. Some differences between quaternion matrices and split quaternion matrices are summarized. A counter example for Gerschgorin theorems for left and right eigenvalues of a split quaternion matrix is given. Finally, a method for finding right eigenvalues of a split quaternion matrix pencil is presented.     

Pushing the envelope of the test functions in the Szegö and Avram–Parter theorems

The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far.     

改进的预处理共轭斜量法及其在工程有限元分析中的应用

本文就预处理共轭斜量法(PCCG法)给出了两个具有理论和实际意义的定理,它们分别讨论了迭代解的定性性质和迭代矩阵的构造原则.作者提出了新的非M-矩阵的不完全LU分解技术和迭代矩阵的构造方法.用此改进的PCCG法,对病态问题和大型三维有限元问题进行了计算并与其他方法作了对比,分析了PCCG法在求解病态方程组时的反常现象.计算结果表明本文建议的方法是求解大型有限元方程组和病态方程组的一种十分有效的方法.     

关于加边矩阵的奇异性问题的注记

在文[1]的基础上,这篇注记给出了m×m复矩阵A的一类非奇异加边矩阵的特征,得到了利用这种加边矩阵的逆阵的子块求全体(1,2)-逆与Moors—Penrose逆所关联的两个定理。 本文约定:C~(m×n)表示m×n复矩阵的集合,C_r~(m×n)表示C~(m×n)的秩r的矩阵的子集,设A∈C~(m×n),通常把Penrose方程