浅谈广义逆矩阵

文章介绍莫尔-潘鲁斯(Moore-Penrose)广义逆矩阵的概念及其与实际背景的联系。文章中定理1和定理2说明条件I与相容线性方程组的基本解的广义逆矩阵的联系,定理3说明条件I和IV与相容线性方程组的最小模解的广义逆矩阵的联系。     

耦合Sylvester矩阵方程的梯度迭代算法

通过推广求解矩阵方程AX=b或AX+XB=C的递推迭代算法和基于递阶辩识原理的思想,给出了求解广义耦合矩阵方程的梯度迭代算法。并证明了迭代算法的收敛性。分析表明,若矩阵方程有唯一解,则对任意的初始值该算法给出的迭代解都能快速的收敛到其精确解。数值实例验证了该算法的有效性。     

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。     

经济核算类矩阵调整的JSD方法及扩展研究

现有的矩阵更新、平衡等调整方法多存在两种缺陷：一是度量新旧矩阵间差异的函数形式不对称,并非严格“距离”概念;二是要求矩阵元素非负导致使用范围受限。为改进上述不足,本文提出了包括Jensen–Shannon divergence（JSD）在内的若干基于对称距离优化的新方法,并统一进行保号、误差妥协等扩展以方便实际运用。在此基础上本文利用中国及其他28个国家的数据,对比分析了多种矩阵调整方法的实际效果,结果发现：第Ⅰ类JSD方法表现最突出且相对稳健,值得代替现在使用较广泛的RAS或交叉熵方法。     

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。     

设计克莱姆法则计算程序的三种思想及C程序

在本文中,根据克莱姆法则讨论其计算程序的三种设计思想,引进具有n行n+1列的增广矩阵的列余子式与列代数余子式的概念,并给出完整的C语言计算程序。在程序中,克莱姆法则中的Dj是通过计算方程组的增广矩阵的第j列的列代数余子式得到的。     

室外大屏幕LED矩阵显示系统的抗干扰设计

室外大屏幕LED矩阵显示系统的抗干扰设计采用了7个抗干扰、低功耗的PIC16F1937单片机组成级联电路,具有恒流输出的TB62726型16位移位锁存器和HF3009恒流驱动电路构建了行、列双路恒流的LED矩阵驱动电路单元。并通过防静电、抗电磁干扰能力的设计,使大屏幕LED显示系统运行可靠。     

矩阵方程的变换解法

本文针对求解矩阵方程的问题,提出了三种不同层次的解法。尤其对利用初等变换解矩阵方程的方法,进行了详细的分析。     

浅谈整数矩阵及其填充问题

文章证明了如果一个部分整数矩阵有一条自由对角线,那么这个矩阵能被填充为一个单模矩阵。这样一个条件从一般意义上讲也是必要的。随后证明了如果一个n×n(n叟2)部分整数矩阵有2n-3个确定的元素并且这些元素中任何n个不构成一行或一列,那么这个矩阵能被填充为一个单模矩阵,这个结果改进了詹的一个最近的结果。     

局部对角占优矩阵的一个性质

引进了局部(α,β,γ)-对角占优矩阵的相关概念,在不可约局部(α,β,γ)-对角占优矩阵的条件下,获得了非奇异H-矩阵及非奇异M-矩阵的实用判别准则,推广了已有的相关结果。     

Towards commercial and environmental excellence: A green portfolio matrix

This article examines the strategic framework and options available for product strategy towards commercial and environmental excellence. It defines the environmental dimension as a ‘moving target’, explores a conceptual framework to integrate commercial and environmental dimensions and elaborates a ‘green portfolio matrix’. The proposed matrix is also empirically investigated as to its practical validity by using a set of secondary data including twelve product cases. Using the matrix, a company can diagnose its existing product mix from an environmental point of view, and furthermore can establish future ‘targets’ for an environmentally-friendly product portfolio. In addition, the matrix can be used as a framework for comparative analyses between strategic business units or for longitudinal assessment of a specific business entity.     

空间加权矩阵设置与空间面板参数估计效率

总结了常用的空间加权矩阵的一般构建方法和研究领域内新提出的空间加权矩阵的构建方法,从宏观与微观层面,量化分析了空间加权矩阵设置对于空间面板参数估计效率、空间效应识别的影响效应。结论表明：宏观数据层面,随着空间加权矩阵复杂程度的提高,无论是空间面板固定效应模型还是空间面板随机效应模型,参数估计的有效性与一致性都显著提高并且广义矩参数估计方法优于拟极大似然估计方法,复合的空间加权矩阵条件下,拉格朗日乘子检验方法的功效更高;微观数据层面,回归结果表明四种不同类型的空间加权矩阵的设置,对于聚集外部性引致的企业全要素生产率增长的空间边界的识别具有显著影响,复合的空间加权矩阵更有效。     

Bayesian Nash equilibrium and variational inequalities

This paper provides a sufficient condition for the existence and uniqueness of a Bayesian Nash equilibrium by regarding it as a solution of a variational inequality. The payoff gradient of a game is defined as a vector whose component is a partial derivative of each player’s payoff function with respect to the player’s own action. If the Jacobian matrix of the payoff gradient is negative definite for each state, then a Bayesian Nash equilibrium is unique. This result unifies and generalizes the uniqueness of an equilibrium in a complete information game by Rosen (1965) and that in a team by Radner (1962). In a Bayesian game played on a network, the Jacobian matrix of the payoff gradient coincides with the weighted adjacency matrix of the underlying graph.     

More on connections between Wishart and matrix GIG distributions

The paper is devoted to relations between the matrix GIG and Wishart distributions. Our basic tool in the first part is a version of the Matsumoto-Yor property for matrix variables. This approach covers the following issues: the Herz identity for the Bessel function of matrix variate argument, characterization of a class of Wishart matrices and linear transformations of the matrix GIG distribution. The Bayesian Wishart model, studied in the second part, gives an alternative definition of the matrix GIG distribution. Such a model is characterized by linearity of conditional expectations and matrix GIG conditional distribution. It is also extended to Bayesian matrix GIG models, in the framework of which an interesting independence property is proved.     

An integrated approach to the development of continuous simulations

An integrated approach to the development of Forrester-style simulation models is described. The approach incorporates the concept of an interaction matrix to assist in the development of causal loop diagrams and Dynamo flow diagrams. The interaction matrix is derived from the fundamental notions of system dynamics. Premised upon the presumption that a computer-aided procedure for model formulation can expedite, systematize, and operationalize the model formulation process, the integrated approach utilizes the interaction matrix as a data structure within the computer. An algorithm designed to interface with a remote terminal (such as a CRT display) determines the interaction matrix by interrogating a user until sufficient information about the problem of interest has been obtained. This algorithm is also described in the paper. The interrogations both motivate and facilitate the determination of quantities to be included as well as couplings between the quantities. When a quantity or coupling is designated by a user, the algorithm “knows” its identity at the time of user origination. Both algorithm and matrix are illustrated through recourse to a text-book example and the paper concludes with a summarizing discussion of the possible contribution of such an approach.     

Portfolio selection based on semivariance and distance correlation under minimum variance framework

In the minimum variance model, the covariance matrix plays an important role because it measures the risk and relationship of asset returns simultaneously under the normality assumption. However, in practice, the distribution of asset returns is nonnormal and has an obvious fat‐tail nature. In addition, the risk is one‐sided. In this paper, the main objective is to propose a better tool to replace the covariance matrix. The covariance matrix can be decomposed into two parts: a diagonal variance matrix and a square matrix with its elements being the Pearson correlation coefficient. A substitution of the covariance matrix is presented by replacing the variance and Pearson correlation coefficient in the decomposition of the covariance matrix with a semivariance and distance correlation coefficient, respectively. The proposed portfolio optimization strategy is applied to empirical data, and the numerical studies show the strategy performs well.     

The interchangeability of safety stocks and safety lead time

Due to the uncertainty in estimating both the demand for end products and the supply of components from lower levels, buffering techniques should be included before the loading of a material requirement planning (MRP) system. Safety stocks and safety lead time are two techniques of providing buffering for loading. There have been many studies made concerning the determination of the amount of safety stocks and safety lead time. Some guidelines for choosing between safety stocks and safety lead time for dealing with uncertainty in both demand and supply also have been established. Although these two different methods have been used successfully, it has not been documented that using these two methods in a given situation will yield essentially the same results; that is, the interchangeability of these two buffering techniques has not been explored quantitatively.Since the net influence of safety stocks and safety lead time and their quantitative interchangeability are of major interest, an analytical model is proposed for this study. The lead-time offset procedure for components loading are represented by a matrix model that is based on a lot-for-lot lot-sizing technique. This lead-time offset matrix model is the product of the precedence matrix and the fixed-duration matrix. The precedence matrix is formed according to the total requirement factor matrix and the duration matrix is formed by each component process time. Thus, the lead-time offset matrix will generate the starting period of each component.When the lead-time offset procedure is modeled, the net influence of buffering quantity can be analyzed. The planned safety stock that is normally used to accommodate unexpected demand, shortage in supply, and defects from the operation at each process can be combined with demand to form the master production schedule. The revised lead time due to the integration of the safety stocks can be calculated through the lead-time offset model. The safety lead time may extend the component process time as well as overall production lead time if the designated safety lead time is longer than the available slack time in a fixed lead-time loading system.When the proposed lead-time offset model is further examined, it is found that planned safety stocks at the higher level can buffer the fluctuations of lower level components quantity as well as the fluctuations of same level components quantity. Safety stocks can also buffer shortages that are caused by the delay of raw material and manufacturing processes. Thus, safety stocks can be used to buffer unexpected delay time up to certain limits. A planned safety lead time at higher level component process can buffer the fluctuations of lower level components process time, as well as the same level component process time. The safety lead time can be used to produce additional products to meet unexpected excessive demand up to certain limits under the following conditions: 1. The excessive demand is known before the actual processing of the components in the lowest level. 2. The raw material at the lowest level is available.Although safety stocks and safety lead time are interchangeable in terms of the ability to buffer variations in quantity, the conditions for safety lead time are seldom met in actual practices. Thus, the slack time in a fixed lead-time loading system cannot be considered as an effective measure to substitute safety stocks. However, all or part of the delay in manufacturing processes or the supply from the lower level components can be buffered by the safety stock and the MPS will still be met. From this study, it is obvious that the slack time can be reduced when safety stocks are planned for an MRP system. The reduction of fixed lead-time duration will be beneficial to the overall planning and scheduling in MRP systems.     

A SAM for Europe: Problems and Perspectives

This paper examines the conceptual and practical problems of constructing a social accounting matrix for Europe. It involves integrating the accounts of member countries within a complete system of regional accounts. The standard matrix framework first proposed by Stone is modified in two respects. First, functional and geographical transactions are distinguished between in order to simplify the accounting structure. Secondly, supraregional accounts are more formally incorporated so as to include accounts for the rest of the world and EC institutions, as well as to accommodate distribution and other transaction costs involved in moving goods, factors and assets between member countries. The general system of regional accounts can then be aggregated, consolidated or apportioned across member states but this has implications for both harmonizing accounts and classifications and considering the extent to which there is truly a ‘single European market’. Finally, the paper considers the availability of data and presents a preliminary compilation of aggregate accounts for three member countries.     

Graphical analysis and financial classification: A case study

This paper illustrates the use of graphical analysis as a complementary diagnostic tool in financial classification problems. For more than two decades statistical models have been used frequently to understand the information content of multivariate data in the context of financial classification. These statistical classification models can be complemented by the use of computer-generated multidimensional data displays and graphical analysis. We illustrate the use of the scatterplot matrix, which is the simplest and very effective form of graphical analysis on a sample of bankrupt and non-bankrupt firms. The paper demonstrates the complementary nature of the scatterplot matrix for tree-structured classification models.