On solving sparse band systems with three algorithms of the ABS family

In this paper, we examine three algorithms in the ABS family and consider their storage requirements on sparse band systems. It is shown that, when using the implicit Cholesky algorithm on a band matrix with band width 2q+1, onlyq additional vectors are required. Indeed, for any matrix with upper band widthq, onlyq additional vectors are needed. More generally, ifa _kj 0,j>k, then thejth row ofH _i is effectively nonzero ifj>i>k. The arithmetic operations involved in solving a band matrix by this method are dominated by (1/2)n ² q. Special results are obtained forq-band tridiagonal matrices and cyclic band matrices.The implicit Cholesky algorithm may require pivoting if the matrixA does not possess positive-definite principal minors, so two further algorithms were considered that do not require this property. When using the implicit QR algorithm, a matrix with band widthq needs at most 2q additional vectors. Similar results forq-band tridiagonal matrices and cyclic band matrices are obtained.For the symmetric Huang algorithm, a matrix with band widthq requiresq–1 additional vectors. The storage required forq-band tridiagonal matrices and cyclic band matrices are again analyzed.This work was undertaken during the visit of Dr. J. Abaffy to Hatfield Polytechnic, sponsored by SERC Grant No. GR/E-07760.     

Reachable and controllable sets for two-dimensional,linear, discrete-time systems

We consider two-dimensional discrete-time linear systems with constrained controls. We propose a simple polynomial time procedure to give an exact external representation of theN-step reachable set and controllable set. The bounding hyperplanes are explicitly derived in terms of the data of the problem. By using a result in computational geometry, all the calculations are made in polynomial time in contrast to classical methods. The limit case asN is also investigated.     

量纲分析的线性代数方法

把量纲概念与矢量空间联系起来，提出用线性代数进行量纲分析的方法，并举例说明。     

Geometric Algebra in Linear Algebra and Geometry

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudo-Euclidean space to isometries in a pseudo-Euclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudo-Euclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative.     

Borel Subalgebras Redux with Examples from Algebraic and Quantum Groups

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J. 45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.     

向量组线性相关性的教学方法与技巧

向量组线性相关性是线性代数教学中的一项重要内容．由于概念比较抽象、定理难以理解，因此一直是线性代数教学环节中的一项难点．通过对向量组线性相关性的定义以及判断方法进行了形象的描述，建立向量组线性相关性与矩阵、线性方程组之间的关系，有利于学生理解向量组线性相关性的真正内涵与简便求解方法．     

Existence of closed-loop policies for constrained discrete-time linear systems with bounded disturbances

We consider discrete-time linear systems with constraints on both control and state variables and bounded disturbances. We exhibit a closed-form expression of the necessary and sufficient conditions for existence of closed-loop policy. These conditions are linear constraints on the initial state and the bounds on the disturbances, control variables and state variables. A simple illustrative example is presented.     

线性代数教材改革指导思想探析

论述线性代数教材改革的必要性及其指导思想，主张线性代数教材建设应以矩阵为主线，以教育部颁发的硕士研究生人学考试线性代数部分的大纲为纲，结合教学现代化的现状，强调几何与代数相结合，重视代数学思想方法的渗透，化抽象为具体，同时在教学中还要引入数学建模的思想．     

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．     

线性代数是蓝色的——大学非数学专业《线性代数》的课程设计

线性代数不仅是处理多元问题的有力工具,而且具有强烈的思辨性.大学本科教育应该是"泛专业的高等素质教育",作为公共基础课之一的《线性代数》,也应着眼于学生综合科学素质的培养,注意对学生进行思维训练.本文提出线性代数的教学理念:"提出处理多元问题的新要求,沿着多元整合的集成化思路推进,逐步把学生引上线性变换和线性空间的思维平台."为此,本文进行了"五模块、两阶段、三层次"的课程设计,希望对讲授该课程的教师有所裨益.