...TARUN GEHLOTS Introduction to transformations * Matrices can be used to represent many transformations on a grid (such as reflections, rotations, enlargements, stretches and shears). * To find the image of a point P, you multiply the matrix by the position vector of the point. Example: A transformation is represented by the 2 by 2 matrix M = . To find the image of the point (3, 2) under this transformation, you need to find the result of the following matrix multiplication So the coordinates of the image are (8, -1). Example 2: A rectangle has coordinates (1, 1), (4, 1), (4, 3) and (1, 3). Find the coordinates of the image of the rectangle under the transformation represented by the matrix . Solution: You could find the image of each vertex in turn by finding , etc. However, it is more efficient to multiply the transformation matrix by a rectangular matrix containing the coordinates of each vertex: . So the image has coordinates (2, 0), (11, -3), (9, -1) and (0, 2). The diagram below shows the object and the image: Any transformation that can be represented by a 2 by 2 matrix, , is called a linear transformation. 1.1 Transforming the unit square The square with coordinates O(0, 0), I(1, 0), J(0, 1) and K(1, 1) is called the unit square. Suppose we consider the image of this square under a general linear transformation as represented by the matrix : . We therefore can notice the following things: * The origin...
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...III. Matrices Definition III.1 An m × n matrix is a set of numbers arranged in a rectangular array having m rows and n columns. It is written A11 A12 · · · A1n A21 A22 · · · A12 A= . . . .. . . . . . . . Am1 Am2 · · · Amn There are two important special cases. A 1 × n matrix (that is, a matrix with 1 row) is called a row vector. An m × 1 matrix (that is, a matrix with 1 column) is called a column vector. Our convention will be that row indices are always written before column indices. As a memory aid, I think of matrices as being RC (Roman Catholic or rows before columns). §III.1. Matrix Operations Definitions 1. Equality. For any two matrices A and B A = B ⇐⇒ (a) A and B have the same number of rows and the same number of columns and (b) Aij = Bij for all i, j 2. Addition. For any two m × n matrices A and B (A + B)ij = Aij + Bij for all 1 ≤ i ≤ m, 1 ≤ j ≤ n That is, the entry in row i, column j of the matrix A + B is defined to be the sum of the corresponding entries in A and B. Note: The sum A + B is only defined if A and B have the same number of rows and the same number of columns. 3. Scalar multiplication. For any number s and any m × n matrix A (sA)ij = sAij for all 1 ≤ i ≤ m, 1 ≤ j ≤ n For example 2 1 0 2 0 + 3 1 1 2×1+0 = 1 2×0+1 2×2+1 2 = 2×3+1 1 5 7 4. Matrix multiplication. For any m × p matrix A and any p × n matrix B p (AB)ik = j=1 Aij Bjk for all 1 ≤ i ≤ m, 1 ≤ k ≤ n ...
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...PROPERTIES OF MATRICES INDEX adjoint.......................4, 5 algebraic multiplicity .....7 augmented matrix.........3 basis.........................3, 7 cofactor ........................4 coordinate vector ..........9 Cramer's rule................1 determinant...............2, 5 diagonal matrix .............6 diagonalizable...............8 dimension .....................6 dot product ...................8 eigenbasis ....................7 eigenspace...................7 eigenvalue ....................7 eigenvector...................7 geometric multiplicity....7 identity matrix ...............4 image ...........................6 inner product................9 inverse matrix...............5 inverse transformation..4 invertible.......................4 isomorphism.................4 kernal ...........................6 Laplace expansion by minors .....................8 linear independence.....6 linear transformation.....4 lower triangular.............6 norm .......................... 10 nullity............................ 8 orthogonal ................ 7, 9 orthogonal diagonalization ................................ 8 orthogonal projection.... 7 orthonormal.................. 7 orthonormal basis ........ 7 pivot columns............... 7 quadratic form.............. 9 rank.............................. 3 reduced row echelon form ................................ 3 reflection ...................... 8 row operations ............. 3 rref ................................3 similarity .......
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...1.0 Introduction In this era of information technology, civil engineers rely heavily on software to perform their design tasks. Unfortunately, most commercial structural analysis packages are closed-source, which means that the operations that the program performs cannot be inspected by the user. Moreover, such software packages are invariably very pricey, and, hence, are generally not affordable for students and smaller engineering firms. The objective of this design project was to design a structural analysis program that would be free of charge and available to all. This computer program was to be open source and well commented, so that its users could comprehend the operations performed in the analysis of a given structure. To accomplish these objectives, the generalized stiffness method of structural analysis was implemented into a computer algorithm. This algorithm, called “TrussT Structural Analysis”, is a collection of visual basic modules embedded in a Microsoft Excel document using Visual Basic for Applications (VBA). This design report outlines the theory behind TrussT Structural Analysis, as well as the methods by which that theory was implemented into computer algorithms. The first two sections of this report present the theory of the generalized stiffness method of structural analysis and its implementation into a computer algorithm. The following sections present the procedures by which the stiffness method was modified to incorporate the analysis...
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...why such methods work. This type of algorithm is used in all professional software for general eigenvalue calculations today, such as MATLAB's eig function. Prerequisite: Sections 5.2 and 6.4 MATLAB functions used: qr, *, eye, :, tic, toc , for, eig qreigdat, qrbasic, qrshift and randint from Lay's Toolbox Part I. Background. It is not easy to calculate eigenvalues for most matrices. Characteristic polynomials are difficult to compute. Even if you know the characteristic polynomial, algorithms such as Newton's method for finding zeros cannot be depended upon to produce all the zeros with reasonable speed and accuracy. Fortunately, numerical analysts have found an entirely different way to calculate eigenvalues of a matrix A, using the fact that any matrix similar to A has the same eigenvalues. The idea is to create a sequence of matrices similar to A which converges to an upper triangular matrix; if this can be done then the diagonal entries of the limit matrix are the eigenvalues of A. The remarkable discoveries are that the method can be done with great accuracy, and it will converge for almost all matrices. In practice the limit matrix is just block upper triangular, not truly triangular (because only real arithmetic is done), but it is still easy to get the eigenvalues from that. See Note 2 below. The primary reason that modern implementations of this method are efficient and reliable is that a QR factorization can be used to create each new matrix in the sequence. Each QR factorization...
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...TUTORIALS Each student must attend one tutorial per week. Assignments will be given for each tutorial and they must be prepared prior to attendance at the tutorial. Most of the assignments will be divided into two Sections A and B. The problems in Section A are to be prepared for the tutorials; those in Section B are intended for additional practice, and not for discussion, however, if you have difficulties with any of them, then you may seek your tutor’s or lecturer’s assistance. EXAMINATIONS Midterm: 40% Final: 60% Examination Date: October 11 and 12th, 2013 - 2:00 – 4:00 p.m. on both days (tentative) TOPICS TO BE COVERED 1. Sets Sets of numbers; real numbers; integers; rational numbers; natural numbers; irrational numbers; Sets and Subsets: set notation; finite and infinite sets; equality of sets; null sets; subsets; proper subsets; comparability of sets; universal sets; power set; disjoint sets; Venn diagrams. Set Operations: Union; intersection; difference; complement, operations on Comparable sets; algebra of sets; cartesian (cross) product of sets. 2. Relations/Functions Relations; domain and range of a relation; relations as sets of ordered pairs; inverse relations. Functions Mappings; domain and range of a function; equality of functions; one-to-one functions; many-to-one functions; constant functions; into functions; onto functions. 3. Sequences and Series Terms of a sequence; terms of a series; the arithmetic series; the geometric series. 4. Limits/Continuity...
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...PROPERTIES OF MATRICES INDEX adjoint.......................4, 5 algebraic multiplicity .....7 augmented matrix.........3 basis.........................3, 7 cofactor ........................4 coordinate vector ..........9 Cramer's rule................1 determinant...............2, 5 diagonal matrix .............6 diagonalizable...............8 dimension .....................6 dot product ...................8 eigenbasis ....................7 eigenspace...................7 eigenvalue ....................7 eigenvector...................7 geometric multiplicity....7 identity matrix ...............4 image ...........................6 inner product................9 inverse matrix...............5 inverse transformation..4 invertible.......................4 isomorphism.................4 kernal ...........................6 Laplace expansion by minors .....................8 linear independence.....6 linear transformation.....4 lower triangular.............6 norm .......................... 10 nullity............................ 8 orthogonal ................ 7, 9 orthogonal diagonalization ................................ 8 orthogonal projection.... 7 orthonormal.................. 7 orthonormal basis ........ 7 pivot columns............... 7 quadratic form.............. 9 rank.............................. 3 reduced row echelon form ................................ 3 reflection ...................... 8 row operations ............. 3 rref ................................3 similarity .......
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...quantitative techniques provides one with the knowledge and skills needed to solve the problems and the challenges of a fast-paced decisionmaking environment. Managers make decisions on a day to day basis and it is necessary for them to be able to analyze the data so as to be able to make optimal decisions. This module has ten lesson which cover matrix algebra, markov analysis, Linear programming, differentiation, applications of differentiation to cost, revenue and profit functions, integral calculus, inventory models, sampling and estimation theory, hypothesis testing and chi-square tests. iii MODULE OBJECTIVES By the end of the course, the student should be able to:- 1. Perform various operations on matrices matrix algebra, 2. Apply the concept of matrices in solving simultaneous equations, input-output analysis and markov analysis, 3. Formulate and solve Linear programming using the graphical and simplex method 4. Differentiate various functions and apply to cost, revenue and profit functions 5. Apply...
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...Matrix Calculations Using Calculators The calculators most commonly used by Foundation Studies students at UTAR are Casio fx-350MS, fx-570MS, and fx-570ES. Casio fx-350MS has no matrix function and cannot be used to perform matrix calculation. Both Casio fx-570MS and Casio fx-570ES have matrix mode. Casio fx-570ES can display fully a 3×3 matrix with all its 9 elements visible. Casio fx-570MS has only a 2-line display and can only show matrix elements one at a time. Hence the operations for matrix computation are different for these two series of calculators. The general procedures in matrix calculations are as follows: (1) Enter Matrix mode of the calculators. (2) Assign a variable to store the matrix. There are 3 variables available: MatA, MatB, and MatC. (3) Select the dimension or order (1×1 to 3×3) of the matrix. (4) Input the elements of the matrix; the data will be automatically saved in the matrix variable assigned. (5) Exit the matrix input or edit mode by pressing the coloured AC key. (6) Press SHIFT MATRIX or SHIFT MAT (Matrix function is at numeric key 4) to recall the stored matrix and perform matrix calculations as needed. (a) Casio fx-570ES (a) Entering a matrix. 1. 2. 3. 4. Press the MODE key. Select 6:Matrix mode. A display Matrix? will be shown to let us select one of the 3 possible matrix variables allowable. Let us choose 1:MatA by pressing 1. The next display permits us to select the order (m×n) of the matrix MatA. Let us choose 1: 3×3 by pressing 1. An...
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...Andre Louis Cholesky Andre Louis Cholesky was a prestigious member of the French army and a successful mathematician. He studied at the Ecole Polytechnique in France and remained in the upper half of his class upon graduating. Following his studies at the Ecole Polytechnique, Andre Louis Cholesky joined the French army and became a second lieutenant. While in the army, he studied at the school d'Application de l'Artillerie et du Génie where he was once again in the most intellectual part of his class. He spent some time serving his country in Tunisia where the French were able to stimulate the economy and establish modern communication. After serving a mission in Tunisia, he was transferred to Algeria where he served another mission. The French treated their efforts in Algeria completely differently than their efforts in Tunisia. They developed hospitals, medical services, and new communications, but dominantly took control of the country and its native people. Upon leaving Algeria, Cholesky joined the Geodesic section of the army geographic service. It was said that Cholesky had “ a sharp intelligence and a great facility for mathematical work, having an inquiring spirit and original ideas.” This field of the army put his true strengths of mathematical excellence to the test. His works had profound effects on the allocations of continents on the map. Cholesky produced a computational procedure, which was fairly simple compared to the previous works of Jean Baptiste Joseph...
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...it is a 3 x 4 matrix. We denote the element on the second row and fourth column with a2,4. Square matrixIf a matrix A has n rows and n columns then we say it's a square matrix. In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements. Remark. There is no difference between a 1 x 1 matrix and an ordenary number. Diagonal matrix A diagonal matrix is a square matrix with all de non-diagonal elements 0. The diagonal matrix is completely denoted by the diagonal elements. Example. [7 0 0] [0 5 0] [0 0 6] The matrix is denoted by diag(7 , 5 , 6) Row matrix A matrix with one row is called a row matrix Column matrix A matrix with one column is called a column matrix Matrices of the same kindMatrix A and B are of the same kind if and only if A has as many rows as B and A has as many columns as B The tranpose of a matrix The n x m matrix A' is the transpose of the m x n matrix A if and only if The ith row of A = the ith column of A' for (i = 1,2,3,..n) So ai,j = aj,i' The transpose of A is denoted T(A) or AT 0-matrix When all the elements of a matrix A are 0, we call A a 0-matrix. We write shortly 0 for a 0-matrix. An identity matrix I An identity matrix I is a diagonal matrix with all diagonal element = 1. A scalar matrix S A scalar matrix S is a diagonal matrix with all diagonal elements alike. a1,1 = ai,i for (i = 1,2,3,..n) Matrix in mathematics, a system of elements aij (numbers, functions...
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...MATLAB® Getting Started Guide R2011b How to Contact MathWorks Web Newsgroup www.mathworks.com/contact_TS.html Technical Support www.mathworks.com comp.soft-sys.matlab suggest@mathworks.com bugs@mathworks.com doc@mathworks.com service@mathworks.com info@mathworks.com Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 (Phone) 508-647-7001 (Fax) The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 For contact information about worldwide offices, see the MathWorks Web site. MATLAB® Getting Started Guide © COPYRIGHT 1984–2011 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by, for, or through the federal government of the United States. By accepting delivery of the Program or Documentation, the government hereby agrees that this software or documentation qualifies as commercial computer software or commercial computer software documentation as such terms are used or defined in FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014. Accordingly, the terms and conditions of this Agreement and only those rights...
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...Basic Directx Transformation with VB.NET Introduction In this tutorial, we will try to discuss the basic transformation in DirectX. We will discuss only the world transform because it’s the simplest transform, but honestly I should say it’s a complicated subject, especially if you tried to understand the underlying concepts of transformation, then you will be lost in pure math problems. So, I will not discuss the mathematical concepts of vectors and matrices (because I don’t understand it myself!) but we learn how to use them for transformation. Background This tutorial is a continuation of my previous tutorial, “Starting Directx using Visual Basic” so I assume you already understand how to create a DirectX device. Draw a Square “Download incomplete_project and start working on it.” Firstly, we will draw a square before transforming it. Declare this variable in your class: Dim buffer As VertexBuffer In creat_vertxbuffer Sub, write: Sub creat_vertxbuffer() buffer = New VertexBuffer(GetType(CustomVertex.PositionColored), 4, device, _ Usage.None, CustomVertex.PositionColored.Format, Pool.Managed) Dim ver(3) As CustomVertex.PositionColored ver(0) = New CustomVertex.PositionColored(-0.5F, -0.5F, 0, Color.Red.ToArgb) ver(1) = New CustomVertex.PositionColored(-0.5F, 0.5F, 0, Color.Green.ToArgb) ver(2) = New CustomVertex.PositionColored(0.5F, -0.5F, 0, Color.Blue.ToArgb) ver(3) = New CustomVertex.PositionColored(0...
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...1 Eigenvalues And Eigenvectors Aamir Nazir Course:- B.Tech 2nd Year (Civil Engineering) Section:- A Roll No.:- 120107002 System ID:- 2012018068 Subject:- Mathematics Subject Code:- MTH-217 Course Code:- CE-107 Teacher Incharge:- Ms. Archana Prasad 2 Contents 1. Abstract 3 2. Introduction 3-4 3. Eigenvectors and Eigenvalues of a real matrix 4 a. Characteristic Polynomial 7-8 b. Algebraic Multiplicities 8-9 4. Calculation 9 a. Computing Eigenvalues 9 b. Computing Eigen Vectors 10 5. Applications 10 a. Geology and Glaciology 10-11 b. Vibration Analysis 11-12 c. Tensor of Moment of Inertia 12 d. Stress Tensor 12 e. Basic...
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...PHYS3531 Physics Project Multi-valley monolayer materials By Fok Hong Ting 2010280804 Under the supervision of Dr. Wang Yao Content Acknowledgement 4 Introduction 5 Motivation 5 Project outline 5 Chapter I Tight binding model 6 Basics 6 The secular equation 8 Conclusion 9 Chapter II Graphene 11 Formulation 11 π energy band of graphene 15 σ energy bands of graphene 17 Conclusion 18 Chapter III Silicene 19 Tight Binding Hamiltonian of silicene without SOC 20 Constructing orthogonal basis 21 1st order SOC inclusion 24 Conclusion 26 Chapter IV Edge state of Group IV elements 27 Formulation 27 Graphene edge 29 Silicene edge 32 Conclusion 42 References 43 Appendix 44 List of Figures 44 Acknowledgement I hereby would like to express my appreciation and respect to my supervisor Dr. Wang Yao. Although I am not a talented student, Dr. Yao provided me timely support and insight in the field of physics. It is my fortune to take part in this final year project under his guidance. Moreover, I would like to thank Dr. GuiBin Liu and Mr. We Yue for their support and comments. Introduction Motivation One of the most intriguing phenomena in physics is...
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