矩阵教学中观察能力培养探析

矩阵理论的教学是线性代数教学中重要的组成部分,它几乎贯穿线性代数教学的始终.矩阵概念、性质、法则、公式、定理等内容的学习离不开一种心理现象,即对矩阵知识的观察与理解.观察是认识矩阵的基础,亦是矩阵学习过程中逻辑思维形成的触角.在矩阵教学中,强调观察矩阵和行列式的区别与联系及二者不同的应用领域、观察矩阵秩和逆的常见求法、观察考研矩阵证明题的常见题型等意义重大.为此,矩阵教学中观察能力的培养秉承五性,即秉承观察的习惯性、目的性、方法性、全面性、深刻性尤为重要.     

三对角矩阵逆的估计

1引言 三对角矩阵出现在很多应用中,例如,在求解常系数微分方程的比值问题,三次样条插值等应用中都会遇到三对角矩阵.因此这类矩阵非常重要,而且也有很多学者致力于这类矩阵的研究.在一些应用中,比如估计条件数和构造稀疏近似逆预条件子,需要计算三对角矩阵的逆,或者估计其逆元素的界.文献[1-7]给出了关于三对角矩阵逆的一些很好的结果,但是,这些结果大都建立在矩阵对角占优的条件之下,这限制了他们的应用.在本文中,我们给出一种一般三对角矩阵逆元素的估计办法.     

投入占用产出框架下Leontief逆矩阵的变动分析

Leontief逆矩阵是投入产出分析的核心概念。研究直接消耗系数的扰动对Leontief逆矩阵的影响是投入产出分析中结构变动问题的重要议题。本文研究了在投入占用产出框架下,各部门的固定资产消耗对Leontief逆矩阵的变动影响。     

基于Schmidt正交化的快速求逆法

基于Schmidt正交化过程获得了一种计算逆矩阵的新方法.对于可逆矩阵A,有Q=MA,其中Q是酉矩阵,M是下三角矩阵.本文直接从Schmidt规范正交化出发,获得下三角矩阵M的计算公式,从而求得逆矩阵A-1=QHM=AHMTM.     

三对角矩阵求逆问题的思考——从一道课本习题谈起

矩阵求逆是矩阵代数中的一个重要运算,逆矩阵的获得却困难重重.文中介绍一个三对角形矩阵求逆的新思路,它可以作为求解逆矩阵问题的一个"解题模块",学习线性代数课程的一个"认知结构",也可以作为教育数学的一个案例.     

基于线性算子的广义逆矩阵的几何表示北大核心

广义逆矩阵是矩阵理论的重要内容.由于广义逆矩阵的定义众多,计算较为繁杂,使得初学者很难理解和掌握其本质.基于线性方程组求解问题的等价表示,从线性算子的角度展示多种广义逆矩阵定义的背景及其几何直观意义.通过对一个特殊算例的分析与求解,实现了对多种广义逆矩阵的几何解释及其在线性方程组求解中的作用,淡化了广义逆矩阵计算的繁杂,加深初学者对广义逆矩阵的理解与掌握.     

基于线性算子的广义逆矩阵的几何表示

广义逆矩阵是矩阵理论的重要内容.由于广义逆矩阵的定义众多,计算较为繁杂,使得初学者很难理解和掌握其本质.基于线性方程组求解问题的等价表示,从线性算子的角度展示多种广义逆矩阵定义的背景及其几何直观意义.通过对一个特殊算例的分析与求解,实现了对多种广义逆矩阵的几何解释及其在线性方程组求解中的作用,淡化了广义逆矩阵计算的繁杂,加深初学者对广义逆矩阵的理解与掌握.     

基于部分基变量的LP问题矩阵算法

基于部分基变量提出了LP问题的矩阵算法. 该算法以最优基矩阵的一个充分必要条件为基础,首先将一个初始矩阵转化为右端项和检验数均满足要求的矩阵,再转为检验数满足要求的基矩阵,最后转化为最优基矩阵.该算法具有使用范围广、计算规模小、计算过程简化、计算机易于实现的优势.矩阵算法的核心运算是求逆矩阵的运算,提出了矩阵算法的求逆问题,讨论并给出了求逆快速算法,该算法充分利用了矩阵算法迭代过程中提供的原来的逆矩阵的信息经过简单的变换得到新的逆矩阵,该算法比直接求逆法计算效率更高.     

从线性映射的视角学习逆矩阵

逆矩阵概念与线性映射的逆映射概念紧密相关.从线性映射的逆映射这个视角引入可逆矩阵概念对于理解可逆矩阵的本质和逆矩阵的求法具有重要的意义.     

三对角逆M矩阵的判定

1、引言 三对角逆M矩阵是指同时为三对角矩阵和逆M矩阵的一类特殊矩阵．文用图论方法探讨三对角逆M矩阵结构,给出了三对角矩阵为逆M矩阵的充分必要条件．此条件提供了判定三对角矩阵是逆M矩阵的方法,但较复杂．文讨论了这类矩阵在Hadamard积下的封闭性．由于三对角逆M矩阵在理论和应用上都有一定价值,所以,寻求一种简单而实用的判定方法是必要的．本文通过对这类矩阵结构特点的研究找到了这样一种方法．同时,由此证明了这类矩阵在Hadamard积下的封闭性．     

可逆矩阵在保密通信中的应用

作为工科“线性代数”课中相关知识的一个具体应用的例子,从理论与实践相结合的角度论述了可逆矩阵在保密通信中的应用及其存在的问题与对策等.     

An application of the Vandermonde determinant

Eigenvalue is an important concept in Linear Algebra. It is well known that the eigenvectors corresponding to different eigenvalues of a square matrix are linear independent. In most of the existing textbooks, this result is proven using mathematical induction. In this note, a new proof using Vandermonde determinant is given. It is shown that this method is much simpler.     

关于化矩阵为标准形时可逆矩阵求法的探讨

可逆矩阵在矩阵理论和应用中都起着十分重要的作用,本文就化矩阵为标准形时可逆矩阵的求法问题进行探讨,给出了利用矩阵的初等行、列变换同时求出两个可逆矩阵的一种简便实用的方法.     

Nonlinear learning of linear algebra: active learning through journal writing

Students find difficulty in learning linear algebra because of the abstraction and formalism associated with concepts such as vector space, linear independence, rank and invertible matrices. Learning the necessary procedures becomes insufficient, and imitating worked examples does not guarantee the maturity level necessary for understanding these concepts. Instructors who seriously consider education reform look for teaching modes that induce active learning; the author resorted to journal writing in the particular context of coordinating between the different definitions and theorems related to the invertibility of a matrix: students were required to write about the various modes of representing invertible matrices, trying to guide them so as not to confuse the object with its representation, a notion referred to by Dorier as cognitive flexibility. As an auxiliary result, learners were observed in the process of building the construct of invertible matrices. The topic of invertible matrices was chosen because it appears that it weaves through the material, and plays the role of a unifier concept. In their final draft, many journals culminated into a student personal paper, entitled ‘Many ways to show invertibility: how to choose the most fit method?' Above all, the project helped students reclaim their intuition and common sense about mathematics.     

More results on Schur complements in Euclidean Jordan algebras

In a recent article Gowda and Sznajder (Linear Algebra Appl 432:1553–1559, 2010) studied the concept of Schur complement in Euclidean Jordan algebras and described Schur determinantal and Haynsworth inertia formulas. In this article, we establish some more results on the Schur complement. Specifically, we prove, in the setting of Euclidean Jordan algebras, an analogue of the Crabtree-Haynsworth quotient formula and show that any Schur complement of a strictly diagonally dominant element is strictly diagonally dominant. We also introduce the concept of Schur product of a real symmetric matrix and an element of a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. An Oppenheim type inequality is proved in this setting.     

关于稳定可逆矩阵的分解(英文)

证明了稳定可逆矩阵相似于两个循环矩阵的积.并且进一步地证明了稳定可逆矩阵可表成对角矩阵与换位子的积.     

On some subclasses of P‐matrices

A matrix with positive row sums and all its off‐diagonal elements bounded above by their corresponding row averages is called a B‐matrix by J. M. Peña in References (SIAM J. Matrix Anal. Appl. 2001; 22 :1027–1037) and (Numer. Math. 2003; 95 :337–345). In this paper, it is generalized to more extended matrices—MB‐matrices, which is proved to be a subclass of the class of P‐matrices. Subsequently, we establish relationships between defined and some already known subclasses of P‐matrices (see, References SIAM J. Matrix Anal. Appl. 2000; 21 :67–78; Linear Algebra Appl. 2004; 393 :353–364; Linear Algebra Appl. 1995; 225 :117–123). As an application, some subclasses of P‐matrices are used to localize the real eigenvalues of a real matrix. Copyright © 2007 John Wiley & Sons, Ltd.     

《线性代数》内容的关联性研究

《线性代数》学科知识点关联性强、相互交叉相互引证的情况比较多.清晰把握知识点之间的关系成为有效教学的重要前提.本文通过建构知识点关联层次图,试图理顺知识点关系、把握好内容格局,并结合教学经验提出一种教学时序.