线性方程组的同解变换与初等变换

本文证明了初等变换是线性方程组仅有的同解变换。 定义1 两个线性方程组若有相同的解集,则称它们是同解的。 定义2 线性方程组的初等变换指的是对线性方程组施行的以下变换:     

线性方程组解的有关问题

首先讨论了两个齐次线性方程组有非零公共解的充分必要条件并给出了非零公共解的一般形式,然后讨论了两个线性方程组同解的一个充分必要条件和非齐次线性方程组的线性无关解向量的个数以及非齐次线性方程组通解的表达式,最后证明了非齐次线性方程组有解的一个充分必要条件.     

关于两个线性方程组同解条件的再思考

首先给出了两个线性方程组Ax=c及Bx=d的解与解之间的关系,通过对两个方程组有公共解的条件的研究,从而给出了两个方程组有同解的充分必要条件.根据所得结论,最后给出了两个线性方程组是否有同解的判别方法以及同解的求解方法.     

非齐次线性方程组同解定理

非齐次线性方程组同解定理王冰，任化民（大连市６２中学）（大阵陆军学院）无论解斤次线性方程组，还是解非万次线性方程组，所用的方法多见消元法。即对其系数矩阵式增广矩阵施以行的初等变换，而得到比较简单的同解方程组。扭矩阵理论来说，就是劳烈对附。技增厂“双阵...     

非齐次线性代数方程组的同解定理

非齐次线性代数方程组的同解定理张会勇，缪光淮，任化民（大连陆军学院）无论解齐次线性方程组，还是解非齐次线性方程组，所用的基本上都是消无法。也就是对其系数矩阵或增度矩阵施以行的初等变换，而得到比较简单的同解方程组。用矩阵理论来说，就是系数矩阵或增度矩阵...     

基于线性方程组同解证明矩阵秩相等的方法

本文探讨了基于线性方程组同解证明矩阵秩相等的方法.     

非齐次线性方程组同解定理

本文从矩阵理论出发,对所谓的非齐次线性方程组同解定理,给出证明。     

向量组关系与其构造的齐次线性方程组的解关系探讨

本定义了一个向量组所能构造的两个齐次线性方程组：分量型和向量型齐次线性方程组．随后得出：向量组之同的线性表示和等价性与它们对应的分量型齐次线性方程组的解之间关系不大,但与它们对应的向量型齐次线性方程组的解之同的关系密切,可以相互刻射．在此基础上指出了教材《线性代数》(第二版,同济大学数学教研室编)中有关部分的安排及叙述的不妥之处．     

直接构造基础解系或通解的线性方程组解法

给出的求解线性方程组基础解系或通解的方法是一种直接构造的方法,不用回到同解方程组,而是通过初等行变换一气呵成,直接写出基础解系或通解.方法简单,易于学生掌握.     

解线性方程组可做列变换

本文介绍一种解线性方程组时可以做列初等变换的方法,并给出A、B∈R_(m×n)时,AX=0与BX=0同解的等价条件。     

Global optimization for sum of geometric fractional functions

This paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints, which arise in various practical problems. By using an equivalent transformation and a new linear relaxation technique, a linear relaxation programming problem of the equivalent problem is obtained. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems. Numerical results are reported to show the feasibility of our algorithm.     

Projective Invariants of Planar 2-Webs

In this paper, we compute the rational projective differential invariants of linear planar 2- webs. The invariants are used to find conditions under which two linear planar 2-webs are equivalent with respect to the group of projective transformations.     

Equivalent linear observer-based tracker for stochastic chaotic system with delays and disturbances

Based on the state-space self-tuning control methodology, aunified method for determining the equivalent linear observer-basedtracker for a stochastic chaotic system with input and statedelays and deterministic disturbances is developed. In thisapproach, an equivalent linear observer is obtained for theconcerned complex system, which may have unknown system parameters,system and measurement noises, and inaccessible system states.More precisely, an equivalent delay-free system is obtainedin the estimating process of the self-tuning control loop, andthen a linear controller and an observer are designed. The proposedmethod significantly simplifies the design and the implementationprocedures of the tracker.     

Controllability Properties of Linear Mean-Field Stochastic Systems

We study some controllability properties for linear stochastic systems of mean-field type. First, we give necessary and sufficient criteria for exact terminal-controllability. Second, we characterize the approximate and approximate null-controllability via duality techniques. Using Riccati equations associated to linear quadratic problems in the control of mean-field systems, we provide a (conditional) viability criterion for approximate null-controllability. In the classical diffusion framework, approximate and approximate null-controllability are equivalent. This is no longer the case for mean-field systems. We provide sufficient (algebraic) invariance conditions implying approximate null-controllability. We also present a general class of systems for which our criterion is equivalent to approximate null-controllability property. We also introduce some rank conditions under which approximate and approximate null-controllability are equivalent. Several examples and counter-examples as well as a partial algorithm are provided.     

Elementary Equivalence of Linear Groups Over Rings with a Finite Number of Central Idempotents and Over Boolean Rings

In the present paper, we start with a criterion of elementary equivalence of linear groups over rings with just a finite number of central idempotents. Then we study elementary equivalence of linear groups over Boolean algebras. We prove that two linear groups over Boolean algebras are elementarily equivalent if and only if their dimensions coincide and these Boolean algebras are elementarily equivalent.     

Synthesis of linear systems with desired equivalent form

A method for feedback synthesis of linear control systems with desired linearly equivalent form of the closed loop system matrix is proposed. The method is based on the serial canonical form of linear multivariable systems which is an alternative to the Luenberger canonical form. A stable computational algorithm for finding the serial canonical form using orthogonal similarity transformations is described. The algorithm for synthesis involves a simple recurrent procedure and gives the possibility to achieve any attainable equivalent form of the closed loop system. The results obtained are extended to the synthesis of reduced order state observers.     

Stabilizability of linear switching systems

A complete characterization of stabilizability for linear switching systems is not available in the literature. In this paper, we show that the asymptotic stabilizability of linear switching systems is equivalent to the existence of a hybrid Lyapunov function for the controlled system, for a suitable control strategy. Further, we prove that asymptotic stabilizability of a switching system with minimum dwell time, is equivalent to Input to State Stability (ISS) of the controlled switching system, with a stabilizing control law. We then derive some structural reductions of the hybrid state space, which allow a decomposition of the original problem into simpler subproblems. The relationships between this approach and the well-known Kalman decomposition of linear dynamic control systems are explored.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.     

The equivalence of linear codes implies semi-linear equivalence

Designs, Codes and Cryptography - We prove that if two linear codes are equivalent then they are semi-linearly equivalent. We also prove that if two additive MDS codes over a field are equivalent...     

Sequential and simultaneous aggregation of diophantine equations

For any system of linear algebraic equations with integer coefficients and a bounded set of nonnegative integer solutions, an infinite number of equations exist, each of which has the same set of nonnegative integer solutions as the given system. Two approaches (sequential and simultaneous aggregation) to the problem of finding the latter equivalent equation are considered. New procedures for sequential and simultaneous aggregation are presented, which improve known results, i.e., provide equivalent equations with smaller coefficients.