Stress Matrices and M Matrices

MatrixEponenential-指数矩阵计算

is invertible then .

symmetric, and that if X is skew-symmetric then e X is orthogonal. exp(X*) = (e X)*, where X* denotes the conjugate transpose of X. It follows that if X is Hermitian then e X is also Hermitian, and that if X is skew-Hermitian then e X is unitary. Linear differential equations One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations. Indeed, it follows from equation (1) below that the solution of where A is a matrix, is given by The matrix exponential can also be used to solve the inhomogeneous equation See the section on applications below for examples. There is no closed-form solution for differential equations of the form where A is not constant, but the Magnus series gives the solution as an infinite sum. The exponential of sums We know that the exponential function satisfies e x + y = e x e y for any numbers x and y. The same goes for commuting matrices: If the matrices X and Y commute (meaning that XY = YX), then However, if they do not commute, then the above equality does not necessarily hold. In that case, we can use the Baker-Campbell-Hausdorff formula to compute e X + Y. The exponential map Note that the exponential of a matrix is always a non-singular matrix. The inverse of e X is given by e-X. This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map from the space of all n×n matrices to the general linear group, i.e. the group of all non-singular matrices. In fact, this map is surjective which means that every non-singular matrix can be written as the exponential of some other matrix (for this, it is essential to consider the field C of complex numbers and not R). The matrix logarithm gives an inverse to this map. For any two matrices X and Y, we have

Invasion Assay Using 3D Matrices

Invasion Assay Using 3D Matrices 1. Overview Scientists have developed 3D models to more accurately study cell invasion and migration processes. While most traditional cell culture systems are 2D, cells in our tissues exist within a 3D network of molecules known as the extracellular matrix or ECM. While many of the mechanistic processes required for cell motility in 2D and 3D are similar, factors such as the reduced stiffness of ECM compared to plastic surfaces, the addition of a third dimension for migration, and the physical hindrance of moving through the mesh of long polymers in the ECM, all present different challenges to the cell compared to two-dimensional migration. This video will briefly introduce the basic function and structure of the ECM, as well as the mechanisms by which cells modulate and migrate through it. Next, we’ll discuss a general protocol used to study endothelial cell invasion. Finally, we will highlight several applications of 3D matrices to studying different biological questions. 2. ECM Composition and Cell-ECM Interaction Let’s begin by examining the composition of the ECM, and how cells interact with it.

A metric for covariance matrices

A Metric for Covariance Matrices Wolfgang F¨o rstner and Boudewijn Moonen Diese S¨a tze f¨u hren dahin,die Theorie der krummen Fl¨a chen aus einem neuen Gesichtspunkt zu betrachten,wo sich der Untersuchung ein weites noch ganz unangebautes Feld ¨o ?net ...so begreift man,dass zweierlei wesentlich verschiedene Relationen zu unterscheiden sind,theils nemlich solche,die eine bestimmte Form der Fl¨a che im Raume voraussetzen,theils solche,welche von den verschiedenen Formen ...unabh¨a ngig sind.Die letzteren sind es,wovon hier die Rede ist ...man sieht aber leicht,dass eben dahin die Betrachtung der auf der Fl¨a che construirten Figuren,...,die Verbindung der Punkte durch k¨u rzeste Linien ?]u.dgl.geh¨o rt.Alle solche Untersuchungen m¨u ssen davon ausgehen,dass die Natur der krummen Fl¨a che an sich durch den Ausdruck eines unbestimmten Linearelements in der Form √(Edp 2+2F dpdq +Gdq 2)gegeben ist ... Carl Friedrich Gauss Abstract The paper presents a metric for positive de?nite covariance matrices.It is a natural expression involving traces and joint eigenvalues of the matrices.It is shown to be the distance coming from a canonical invariant Riemannian metric on the space Sym +(n,R )of real symmetric positive de?nite matrices In contrast to known measures,collected e.g.in Grafarend 1972,the metric is invariant under a?ne transformations and inversion.It can be used for evaluating covariance matrices or for optimization of measurement designs. Keywords:Covariance matrices,metric,Lie groups,Riemannian manifolds,exponential map-ping,symmetric spaces 1Background The optimization of geodetic networks is a classical problem that has gained large attention in the 70s. 1972E.W.Grafarend put together the current knowledge of network design,datum transfor-mations and arti?cial covariance matrices using covariance functions in his classical monograph [?];see also [?].One critical part was the development of a suitable measure for comparing two covariance matrices.Grafarend listed a dozen measures.Assuming a completely isotropic network,represented by a unit matrix as covariance matrix,the measures depended on the eigenvalues of the covariance matrix. 1983,11years later,at the Aalborg workshop on ’Survey Control Networks’Schmidt [?]used these measures for ?nding optimal networks.The visualization of the error ellipses for a single point,leading to the same deviation from an ideal covariance structure revealed de?ciencies of these measures,as e.g.the trace of the eigenvalues of the covariance matrix as quality measure ?]emphasized by the authors

Raven’s Progressive Matrices

Raven’s Progressive Matrices Tamara M. Burns, M.Ed. Deborah Mazur, M.Ed. Erich R. Merkle, M.A., M.Ed. Cognitive Assessment October 30, 2000

Raven’s Progressive Matrices Presentation Outline: –Description of Test (Tamara) –Psychometric Info & Critique (Erich) –Demonstration (Erich) –Practical Uses & Concerns (Debbie) –Any Questions (All)

Raven’s Progressive Matrices ?By J.C. Raven, J.H. Court and J. Raven ?Published by Oxford Psychologists Press, Ltd. ?Originally Introduced in 1938 ?Most recent version was published in 1995

Description of Test ?Non-verbal test of reasoning ability based on figural test stimuli ?Measures the ability to make comparisons, to reason by analogy and to organize spatial perceptions into systematically related wholes

矩阵应用简介

矩阵应用简介 The introduction of Matrix application 作者：刁士琦 2015/12/27

摘要 本课题以线性代数的应用为研究对象，通过网络、书籍查询相关知识与技术发展。 全文分为四部分，第一部分是绪论，介绍本课题的重要意义。第二部分是线性代数的发展。第三部分是经典矩阵应用。第四部分是矩阵应用示例。第五部分为结论。 关键词：莱斯利矩阵模型、希尔密码

目录 摘要 (2) 1 引言 (4) 2 矩阵的发展 ............................................................................................ 错误！未定义书签。 3 经典矩阵应用 (4) 3.1矩阵在经济学中的应用 (4) 3.2矩阵在密码学中的应用 (7) 3.3莱斯利矩阵模型 (5) 4 矩阵应用示例 (6) 4.1经济学应用示例 (6) 4.2希尔密码应用示例 (7) 4.3植物基因分布 (7) 6 结论 (8) 参考文献 (9)

1引言 线性代数是以向量和矩阵为对象，以实向量空间为背景的一种抽象数学工具，它的应用遍及科学技术的国民经济各个领域。 2矩阵的发展 1850年，西尔维斯特在研究方程的个数与未知量的个数不相同的线性方程时，由于无法使用行列式，所以引入了Matrix-矩阵这一词语。现代的矩阵理论给出矩阵的定义就是：由mn 个数排成的m行n列的数表。在此之后，西尔维斯特还分别引入了初等因子、不变因子的概念[5]。虽然后来一些著名的数学家都对矩阵中的不同概念给出了的定义，也在矩阵领域的研究中做了很多重要的工作。但是直到凯莱在研究线性变化的不变量时，才把矩阵作为一个独立的数学概念出来，矩阵才作为一个独立的理论加以研究。 矩阵概念的引入，首先是由凯莱发表的一系列和矩阵相关的文章，将零散的矩阵的知识发展为系统完善的理论体系。矩阵论的创立应归功与凯莱。凯莱在矩阵的创立过程中做了极大的贡献。其中矩阵的转置矩阵、对称矩阵和斜对称矩阵的定义都是由凯莱给出的。“从逻辑上来说，矩阵的概念应限于行列式的概念，但在历史上却正好相反。”凯莱如是说。1858年，《A memoir on the theory of matrices》系统阐述了矩阵的理论体系，并在文中给出了矩阵乘积的定义。 对矩阵的研究并没有因为矩阵论的产生而停止。1884年，西尔维斯特给出了矩阵中的对角矩阵和数量矩阵的定义。1861年，史密斯给出齐次方程组的解的存在性和个数时引进了增广矩阵和非增广矩阵的术语。同时，德国数学家弗罗伯纽斯的贡献也是不可磨灭的，他的贡献主要是在矩阵的特征方程、特征根、矩阵的秩、正交矩阵、矩阵方程等方面。并给出了正交矩阵、相似矩阵和合同矩阵的概念，指明了不同类型矩阵之间的关系和矩阵之间的重要性质。 3经典矩阵应用 3.1矩阵在经济学中的应用 投入产出综合平衡模型是一种宏观的经济模型,这是用来全面分析某个经济系统内

matrix基本使用方法(1)

旋转 围绕点px, py 旋转 degrees度, 如果没设置坐标,默认以0,0点旋转.

例子: setRotate(45, 180, 120); 缩放,翻转 以点px,py为原点缩放 >=0 1为正常大小 如果是负数,图形就会翻转 如果没设置原点坐标,默认以0,0点缩放(如果发现图片不见了,检查一下是不是翻转出了屏幕) 例子:setScale(-0.5f, 1,180, 120); //左右翻转并缩放到一半大小

倾斜 以点px,py为原点倾斜如果没有设置原点,则以0,0点为原点. 例子:setSkew(0, 1, 180, 120); //Y 方向拉伸

坐标 是图片移动到某一个位置 注意 Matrix中带有pre, post的函数需要考虑先后顺序 例如:想要旋转45度,然后平移到100,100的位置需要

Java代码 1.Matrix matrix = new Matrix(); 2.matrix.postRotate(45); 3.matrix.postTranslate(100, 100); 或者 Java代码 1.Matrix matrix = new Matrix(); 2.matrix.setTranslate(100, 100); 3.matrix.preRotate(45); 这就要考虑到矩阵的前乘和后乘了,不然的话你会发现可能坐标位置不是你想要的,可能图像都不见了. 如果在复杂一些,需要选择,缩放,倾斜同时起作用,并且还要设置坐标到屏幕指定位置你会发现很麻烦,需要自己计算出各个方法的参数,然后考虑调用的先后顺序. 但这里有一种更简便的方法,叫系统帮我们计算 这个方法的意思是帮我们把两个 Matrix对象计算并连接起来. 这样的话我们就可以这样使用了

矩阵转换表示(翻译)

This article copy form MSDN 2001. mk:@MSITStore:C:\Program%20Files\Microsoft%20Visual%20Studio\MSDN\2001OCT\103 3\gdicpp.chm::/hh/gdicpp/cpp_aboutgdip05_00c4.htm Platform SDK: GDI+ Translate By guozhengkun 转换矩阵表示 一个m×n matrix is a set of numbers arranged in m rows and n columns. The following illustration shows several matrices. 一个m×n矩阵是m行n列的一组数字集合。下图列出了几种矩阵。 You can add two matrices of the same size by adding individual elements. The following illustration shows two examples of matrix addition. 你可以通过矩阵每个元素相加实现两个大小相同的矩阵的加法运算 An m×n matrix can be multiplied by an n×p matrix, and the result is an m×p matrix. The number of columns in the first matrix must be the same as the number of rows in the second matrix. For example, a 4×2 matrix can be multiplied by a 2×3 matrix to produce a 4×3 matrix. 一个m×n矩阵乘以n×p矩阵, 结果是m×p矩阵. 第一个矩阵的列数必须和第二个矩阵的行数相同才能相乘。 举例，一个 4×2 矩阵可以通过一个 4×3 矩阵乘以 2×3 矩阵得到。 Points in the plane and rows and columns of a matrix can be thought of as vectors. For example, (2, 5) is a vector with two components, and (3, 7, 1) is a vector with three components. The dot product of two vectors is defined as follows: 平面上的点集和矩阵的行列可以看作向量的集合。例如，(2, 5) 是一个包含两个分量的向量，and (3, 7, 1) 是一个包含三个分量的向量。两个向量的点积定义如下： (a, b) ? (c, d) = ac + bd (a, b, c) ? (d, e, f) = ad + be + cf For example, the dot product of (2, 3) and (5, 4) is (2)(5) + (3)(4) = 22. The dot product of (2, 5, 1) and (4, 3, 1) is (2)(4) + (5)(3) + (1)(1) = 24. Note that the dot product of two vectors is a number, not another vector. Also note that you can calculate the dot product only if the two vectors have the same number of components. 例如， (2, 3) 和 (5, 4) 的点积等于 (2)*(5) + (3)*(4) = 22。 (2, 5, 1) 和 (4, 3, 1) 等于 (2)(4) + (5)(3) + (1)(1) = 24。注意两个向量的点积是一个数值，不是另一个向量。另外，注意只有分量个数相同的 向量才能计算点积。

the square matrices

M¨o ller’s Algorithm Teo Mora(theomora@disi.unige.it) Duality was introduced in Commutative Algebra in1982by the seminal paper[14]but the relevance of this result became clear after –the same duality exposed in[14]was indipendently applied in[5,28] to produce an algorithm for solving any squarefree0-dimensional ideal I?K[X1,...,X n]and –the algorithm developed in[14]was improved in[18]and applied in order to solve the FGLM-problem; –the ideas of[14]and[18]were merged in[26](see also[22])propos-ing an algorithm which produces the Gr¨o bner basis of an a?ne ideal I=∩r i=1q i?K[X1,...,X n],where each q i is a primary ideal at an algebraic point,equivalently given by its inverse system,or Gr¨o bner basis,or even any basis(see[27]). This led to formalize under the label of M¨o ller’s Algorithm[2],the algo-rithm proposed in[14,19,18,26]which solves the following Problem1.Let ?P:=k[X1,...,X n]the polynomial ring over a?eld k, ?T:={X a11···X a n n:(a1,...,a n)∈N n}, ?P?the P-module of the k-linear functionals over P. Given a?nite set L={ 1,..., s}?P?of linearly independent k-linear functionals such that I:={f∈P: i(f)=0,?i}is a zero-dimensional ideal and a term-ordering<,to compute ?the Gr¨o bner basis of I wrt<; ?the corresponding Gr¨o bner escalier N<(I)?T; ?a set q:={q1,...,q s}?P which is triangular to L and satis?les Span k(q)=Span k(N<(I))～=P/I; 1

Hotelling

Package‘Hotelling’ February19,2015 Version1.0-2 Date2013-11-06 Title Hotelling's T-squared test and variants Author James M.Curran Maintainer James M.Curran Description A set of R functions and data sets which implements Hotelling's T^2test,and some vari-ants of it.Functions are also included for Aitchison's additive log ratio and centred log ra- tio transformations Depends corpcor License GPL(>=2) URL https://www.sodocs.net/doc/349399040.html,/showperson?firstname=James&surname=Curran NeedsCompilation no Repository CRAN Date/Publication2013-11-0607:11:01 R topics documented: alr (2) bottle.df (3) clr (3) container.df (4) hotelling.stat (5) hotelling.test (6) plot.hotelling.test (8) print.hotelling.test (9) Index10 1

Hermitian matrix

Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose – that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: If the conjugate transpose of a matrix is denoted by , then the Hermitian property can be written concisely as Hermitian matrices can be understood as the complex extension of real symmetric matrices. Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real. Examples For example, Well-known families of Pauli matrices, Gell-Mann matrices and various generalizations are Hermitian. In theoretical physics such Hermitian matrices usually are multiplied by imaginary coefficients,[1][2] which results in skew-Hermitian matrices (see below). Properties The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real. A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of a Hermitian matrix. Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies. It says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix A are real, and that A has n linearly independent eigenvectors. Moreover, it is possible to find an orthonormal basis of C n consisting of n eigenvectors of A. The sum of any two Hermitian matrices is Hermitian, and the inverse of an invertible Hermitian matrix is Hermitian as well. However, the product of two Hermitian matrices A and B will only be Hermitian if they commute, i.e., if AB = BA. Thus A n is Hermitian if A is Hermitian and n is an integer. The Hermitian complex n-by-n matrices do not form a vector space over the complex numbers, since the identity matrix is Hermitian, but is not. However the complex Hermitian matrices do form a vector space over the real numbers. In the 2n2R dimensional vector space of complex n×n matrices, the complex Hermitian matrices form denotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis a subspace of dimension n2. If E jk can be described as follows: for (n matrices) together with the set of matrices of the form for ((n2?n)/2 matrices) and the matrices for ((n2?n)/2 matrices) where denotes the complex number , known as the imaginary unit.

不可约矩阵与几乎可约矩阵的一些组合性质.

不可约矩阵与几乎可约矩阵的一些组合性质 摘要非负矩阵是指元素为非负实数的矩阵，同计算数学，经济数学，概率论，物理，化学等有着密切关系。本论文主要研究非负矩阵的那些仅依赖 于矩阵的0元素的位置，而与元素本身数值无关的性质。本论文从非负矩阵的 基础理论出发,结合图论的有关性质，利用图论与矩阵的关系，来研究不可约矩阵与几乎可约矩阵的1些性质。本论文分为3部分，第1 章是引言部分，第2章阐述了不可约矩阵，不可约矩阵的谱半径，完全不可分 矩阵，几乎可约矩阵，几乎可分矩阵的概念，第3章阐述了不可约矩阵，不可 约矩阵的谱半径，完全不可分矩阵，几乎可约矩阵，几乎可分矩阵的重要定 理，性质以及其证明。关键字不可约矩阵；完全不可分矩阵；几乎可约矩阵；几乎可分矩阵；极小强连通图 Abstract Nonnegative Matrices is the matrices whose elements are nonnegative real numbers, and it has close relationship with computer science, economic mathematics, the theorem of probability, physical. This paper mainly research the matrices’ quality with only depends on zero in matrices, but not its own values. This paper main research Combinational quality of Irreducible Matrices and Nearly Reducible Matrices by basic theory of Nonnegative Matrices , quality of graph theory ,and the relationship between graph theory and matrices. This paper includes three parts, the first part is introduction, the second one expounds the concept of irreducible matrices, spectral radius of irreducible, fully indecomposable matrices, nearly reducible matrices, and nearly decomposable matrices. The last one expounds important theories, qualities and proof of irreducible matrices, spectral radius of irreducible, fully indecomposable matrices, nearly reducible matrices, and nearly decomposable matrices. KeywordIrreducible matrices; Fully indecomposable matrices; Nearly Reducible matrices; Nearly decomposable matrices; Minimally strong diagraph.