矩阵初等变换的若干应用及注意事项

矩阵的初等变换在线性代数的学习中有着非常重要的地位,本文对初等变换在解线性方程组、求逆矩阵、解矩阵方程、求矩阵的秩、矩阵列向量组的最大无关组等方面作了详细分析,对教材内容的学习是一个很好的补充。     

对矩阵的初等行变换若干应用的阐述

阐述了矩阵的初等行变换的几点应用。利用矩阵的初等行变换可讨论矩阵的可逆性,并求逆矩阵;利用矩阵的初等行变换,可解矩阵方程;利用矩阵的初等行变换,可求矩阵的秩及矩阵的最高阶非零子式;利用矩阵的初等行变换,可讨论线性方程组解的存在性,并求解线性方程组;利用矩阵的初等行变换,可讨论向量是否可由一个向量组线性表示;利用矩阵的初等行变换,可讨论向量组的线性相关性;利用矩阵的初等行变换,可求向量组的秩及其极大无关组。     

齐次线性方程组解的几点应用

应用齐次线性方程组有非零解,只有零解的理论,对四种不同的情况证明矩阵的秩的关系,以及矩阵间的线性关系。     

一类与工程设计相关的Jacobi矩阵逆特征值问题

Jacobi矩阵逆特征值问题,在振动工程、结构设计、工程设计应用和系数参数识别等领域有重要应用,文章考虑了由混合型特征对构造一个Jacobi矩阵的问题,给出了问题有唯一解的充分必要条件。     

矩阵与线性方程组的解

本文分析和阐明了高等代数中用矩阵解线性方程组的问题,同时也简单地概括总结了n阶行列式的计算方法。     

矩阵的初等变换课堂教学设计探讨

矩阵初等变换是求矩阵的秩，求向量组的极大无关组，解线性方程组，求多项式的最大公因式等的关键，而学生往往掌握的不好。从中学所学消元法解方程组对比讲解矩阵的初等变换，能由浅入深的引导学生正确掌握这一知识。     

一阶齐次线性微分方程组解的一个结论

本文根据微分方程和矩阵代数的有关理论,给出了关于一阶线性方程组x'=A(t)x解的伏朗斯基行列式的结论.     

一种基于相容系数的证据组合方法研究

文章针对证据理论在实际应用中所出现的反直观冲突问题,提出了一种基于证据相容度的证据组合方法。根据证据相容性函数获得证据间相容度矩阵,对证据相容度矩阵归一化得到证据的一致相容性度量,在此基础上定义了证据相容系数的偏离度以表征证据的稳定性,进而联合证据的一致相容性与证据的稳定性获得证据的权系数,最后采用D-S组合规则完成对加权平均证据的组合,增加数据的一致性,降低数据的冲突。实验结果和相关分析表明该方法是合理有效的。     

“二维LP问题的一个直接算法”中定理反例

针对"二维LP问题的一个直接算法"一文中的算法基本定理,给出了两个反例,分别说明其中的引理和定理都是错误的,建立在这些结论基础上的求解一般线性规划问题的代数算法无法求出一般线性规划问题的最优解。数值试验表明,随着方程个数的增加,用他们的方法求得正确解的概率将越来越低。给出了一个二维线性规划问题最优解的性质定理,由该定理可导出文[1]中代数算法有效的条件。     

基于EXCEL的动力总成悬置系统模态及解耦的计算方法与验证

根据动力总成悬置系统模态及解耦的计算公式,利用EXCEL自带的有限的矩阵运算函数,计算动力总成悬置系统的广义刚度矩阵K和广义质量矩阵M的逆矩阵M~(-1)的乘积M~(-1)×K(设其结果为矩阵A)。然后利用作者发明的一种特殊方法在EXCEL中求出该矩阵A的特征值,再利用逆冥法在EXCEL表中求出矩阵A的特征向量,最终计算出动力总成悬置系统的模态及解耦。整个计算过程庞大而复杂,但求解矩阵A的特征值和特征向量是核心,因此将重点介绍。此外,将利用工程领域应用极广的计算软件MATLAB对基于EXCEL的动力总成悬置系统模态及解耦的计算的准确性和精度进行验证。     

Optimal selling mechanisms for multiproduct monopolists: incentive compatibility in the presence of budget constraints

We analyze the nonlinear pricing problem faced by an incomplete information monopolist operating in a market populated by agents with budget constraints. We show that if other goods are available and if the monopolist's goods are nonessential relative to other goods, then there exists an optimal, individually rational, and incentive compatible selling mechanism for the monopolist (Theorem 1). Moreover, we show that a solution to all such nonlinear pricing problems exists if and only if the monopolist's goods are nonessential (Theorem 2). In the absence of nonessentiality, we show that if the monopolist's profit function is independent of quantity (e.g., if all costs are fixed), then an optimal selling mechanism exists (Theorem 3). Finally, we show that if there is reporting (of types by agents) and partial recognition of types (by the monopolist), then an optimal selling mechanism exists, even in the absence of nonessentiality, provided agents' utility functions are affine and continuous in goods (Theorem 4).     

中国多机制门限金融状况指数编制及应用

研究目标：实证编制及应用中国多机制门限金融状况指数(MR-TFCI)。研究方法：通过拓展构建了多机制门限向量自回归(MR-TVAR)模型和MR-TFCI的多机制门限编制公式,从经济增长(RG)目标出发,选取5个金融变量,测算四个机制的广义脉冲响应函数值,编制了中国MR-TFCI,并比较其与 2机制门限FCI(2R-TFCI)和1机制线性FCI(1R-FCI)的优劣。研究发现：与2R-TFCI和1R-FCI相比,MR-TFCI是RG更优的先行、相关性、因果性和预测指标;中国货币政策调控经济增长的效应和传导渠道具有门限特征;中国货币政策调控经济增长的方式类型是价格和数量结合型的。研究创新：构建了MR-TVAR模型和MR-FCI的多机制门限编制公式,编制了首个多机制门限金融状况指数。研究价值：为中国政府部门实施货币政策和实体经济进行投融资决策提供了科学决策依据。     

Generalizing the standard product rule of probability theory and Bayes's Theorem

In this paper the usual product rule of probability theory is generalized by relaxing the assumption that elements of sets are equally likely to be drawn. The need for such a generalization has been noted by Jeffreys [1998. Theory of Probability, 3rd ed, reprinted in Oxford classics Series, Oxford University Press (1st ed.,1939), Oxford. pp. 24–25], among others, in his work on an axiom system for scientific learning from data utilizing Bayes's Theorem. It is shown that by allowing probabilities of elements to be drawn to be different, generalized forms of the product rule and Bayes's Theorem are obtained that reduce to the usual product rule and Bayes's Theorem under certain assumptions that may be satisfactory in many cases encountered in practice in which the principle of insufficient reason is inadequate. Also, in comparing alternative hypotheses, allowing the prior odds to be random rather than fixed provides a useful generalization of the standard posterior odds.     

An invariance property of Farebrother's procedure for estimation with aggregated data

A procedure proposed by Farebrother (1979) for estimating the parameters of a standard Gauss-Markov model from aggregated data is shown to be invariant with respect to the choice of a generalized inverse of the matrix designated to approximate the unknown dispersion matrix of a transformed model, thus correcting Farebrother's statement that this desirable property cannot be attributed to his procedure.     

Stochastic Jacobian and Riccati ODE in affine term structure models

In affine term structure models (ATSM) the stochastic Jacobian under the forward measure plays a crucial role for pricing, as discussed in Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). Their approach leads to a deterministic integro-differential equation which, apparently, has the advantage of by-passing the solution to the Riccati ODE obtained by the standard Feynman-Kac argument. In the generic multi-dimensional case, we find a procedure to reduce such integro-differential equation to a non linear matrix ODE. We prove that its solution does necessarily require the solution of the vector Riccati ODE. This result is obtained proving an extension of the celebrated Radon Lemma, which allows us to highlight a deep relation between the geometry of the Riccati flow and the stochastic calculus of variations for an ATSM. We are grateful to two anonymous referees for their careful reading of the paper.     

识别概念的参数关系说质疑

经济计量学联立方程模型识别概念的参数关系说有两种形式的表述。本文利用线性方程组的求解理论，揭示了参数关系说识别概念的内涵。对于其表述1，我们指出了它叙述的不够准确与不够直观，并在文中给出了相应的建议；对于其表述2，则通过反例证明了它概念上的内在矛盾性。     

Reparameterized inverse gamma regression models with varying precision

In this article, we propose a mean linear regression model where the response variable is inverse gamma distributed using a new parameterization of this distribution that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive response variable, as usual in the context of generalized linear models. The variance function of the proposed model has a quadratic form. The inverse gamma distribution is a member of the exponential family of distributions and has some distributions commonly used for parametric models in survival analysis as special cases. We compare the proposed model to several alternatives and illustrate its advantages and usefulness. With a generalized linear model approach that takes advantage of exponential family properties, we discuss model estimation (by maximum likelihood), black further inferential quantities and diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the obtained results. A real application using minerals data set collected by Department of Mines of the University of Atacama, Chile, is considered to demonstrate the practical potential of the proposed model.     

Single-equation estimators and aggregation restrictions when equations have the same sets of regressors

It is well known that linear equations subject to cross-equation aggregation restrictions can be ‘stacked’ and estimated simultaneously. However, if every equation contains the same set of regressors, a number of single-equation estimation procedures can be employed. The applicability of ordinary least squares is widely recognized but the article demonstrates that the class of applicable estimators is much broaders than OLS. Under specified conditions, the class includes instrumental variables, generalized least squares, ridge regression, two-stage least squares, k-class estimators, and indirect least squares. Transformations of the original equations and other related matters are discussed also.     

On the Relationship between Input–Output Coefficients and Hanoch's Linear Homogeneous Constant Differences of Elasticities of Substitution Production Function

Tilanus's ‘mixed’ input–output coefficients are generalized and it is shown that the generalized coefficients stand in a fixed proportion to each other if and only if the technology is described by Hanoch ‘s linear homogeneous constant differences of elasticities of substitution production function.     

Fisher information under Gaussian quadrature models

This paper develops formulae to compute the Fisher information matrix for the regression parameters of generalized linear models with Gaussian random effects. The Fisher information matrix relies on the estimation of the response variance under the model assumptions. We propose two approaches to estimate the response variance: the first is based on an analytic formula (or a Taylor expansion for cases where we cannot obtain the closed form), and the second is an empirical approximation using the model estimates via the expectation–maximization process. Further, simulations under several response distributions and a real data application involving a factorial experiment are presented and discussed. In terms of standard errors and coverage probabilities for model parameters, the proposed methods turn out to behave more reliably than does the ‘disparity rule’ or direct extraction of results from the generalized linear model fitted in the last expectation–maximization iteration.