We wish to solve a system of N simultaneous linear algebraic equations for the N unknowns x 1, x 2,...,x N, that are expressed in the general form
|
|
- Blake Atkins
- 5 years ago
- Views:
Transcription
1 Linear algebra This chapter discusses the solution of sets of linear algebraic equations and defines basic vector/matrix operations The focus is upon elimination methods such as Gaussian elimination, and the related LU and Cholesky factorizations Following a discussion of these methods, the existence and uniqueness of solutions are considered Example applications include the modeling of a separation system and the solution of a fluid mechanics boundary value problem The latter example introduces the need for sparse-matrix methods and the computational advantages of banded matrices Because linear algebraic systems have, under well-defined conditions, a unique solution, they serve as fundamental building blocks in more-complex algorithms Thus, linear systems are treated here at a high level of detail, as they will be used often throughout the remainder of the text Linear systems of algebraic equations We wish to solve a system of N simultaneous linear algebraic equations for the N unknowns x, x 2,,x N, that are expressed in the general form a x + a 2 x 2 + +a N x N = b a 2 x + a 22 x 2 + +a 2N x N = b 2 () a N x + a N2 x 2 + +a NN x N = b N a ij is the constant coefficient (assumed real) that multiplies the unknown x j in equation i b i is the constant right-hand-side coefficient for equation i, also assumed real As a particular example, consider the system for which x + x 2 + x 3 = 4 2x + x 2 + 3x 3 = 7 (2) 3x + x 2 + 6x 3 = 2 a = a 2 = a 3 = b = 4 a 2 = 2 a 22 = a 23 = 3 b 2 = 7 (3) a 3 = 3 a 32 = a 33 = 6 b 3 = 2 wwwcambridgeorg
2 2 Linear algebra It is common to write linear systems in matrix/vector form as Ax = b (4) where a a 2 a 3 a N a 2 a 22 a 23 a 2N A = a N a N2 a N3 a NN x x 2 x = x N b b 2 b = b N (5) Row i of A contains the values a i, a i2,,a in that are the coefficients multiplying each unknown x, x 2,,x N in equation i Column j contains the coefficients a j, a 2 j,,a Nj that multiply x j in each equation i =, 2,,N Thus, we have the following associations, coefficients multiplying rows equations columns a specific unknown in each equation We often write Ax = b explicitly as a a 2 a N x b a 2 a 22 a 2N x 2 = b 2 a N a N2 a NN x N b N (6) For the example system (2), 4 A = 2 3 b = 7 (7) In MATLAB we solve Ax = b with the single command, x=a\b For the example (2), we compute the solution with the code A=[;23;36]; b = [4; 7; 2]; x=a\b, x= Thus, we are tempted to assume that, as a practical matter, we need to know little about how to solve a linear system, as someone else has figured it out and provided us with this handy linear solver Actually, we shall need to understand the fundamental properties of linear systems in depth to be able to master methods for solving more complex problems, such as sets of nonlinear algebraic equations, ordinary and partial wwwcambridgeorg
3 Review of scalar, vector, and matrix operations 3 differential equations, etc Also, as we shall see, this solver fails for certain common classes of very large systems of equations, and we need to know enough about linear algebra to diagnose such situations and to propose other methods that do work in such instances This chapter therefore contains not only an explanation of how the MATLAB solver is implemented, but also a detailed, fundamental discussion of the properties of linear systems Our discussion is intended only to provide a foundation in linear algebra for the practice of numerical computing, and is continued in Chapter 3 with a discussion of matrix eigenvalue analysis For a broader, more detailed, study of linear algebra, consult Strang (23) or Golub & van Loan (996) Review of scalar, vector, and matrix operations As we use vector notation in our discussion of linear systems, a basic review of the concepts of vectors and matrices is necessary Scalars, real and complex Most often in basic mathematics, we work with scalars, ie, single-valued numbers These may be real, such as 3, 4, 5/7, 3459, or they may be complex, + 2i, /2 i, where i = The set of all real scalars is denoted R The set of all complex scalars we call C For a complex number z C, we write z = a + ib, where a, b Rand a = Re{z} = real part of z b = Im{z} = imaginary part of z (8) The complex conjugate, z = z,of z = a + ib is Note that the product zz is always real and nonnegative, z = z = a ib (9) zz = (a ib)(a + ib) = a 2 iab + iab i 2 b 2 = a 2 + b 2 () so that we may define the real-valued, nonnegative modulus of z, z, as Often, we write complex numbers in polar notation, z = zz = a 2 + b 2 () z = a + ib = z (cos θ + i sin θ) θ = tan (b/a) (2) Using the important Euler formula, a proof of which is found in the supplemental material found at the website that accompanies this book, e iθ = cos θ + i sin θ (3) wwwcambridgeorg
4 4 Linear algebra e [2] = (,,) v v e [] = (,,) v 3 e [3] = (,,) Figure Physical interpretation of a 3-D vector we can write z as z = z e iθ (4) Vector notation and operations We write a three-dimensional (3-D) vector v (Figure ) as v v = (5) v 3 v is real if v,,v 3 R; we then say v R 3 We can easily visualize this vector in 3- D space, defining the three coordinate basis vectors in the (x), 2(y), and 3(z) directions as to write v R 3 as e [] = e [2] = e [3] = (6) v = v e [] + e [2] + v 3 e [3] (7) We extend this notation to define R N, the set of N-dimensional real vectors, v v = v N (8) where v j Rfor j =, 2,,N By writing v in this manner, we define a column vector; however, v can also be written as a row vector, v = [v v N ] (9) The difference between column and row vectors only becomes significant when we start combining them in equations with matrices wwwcambridgeorg
5 Review of scalar, vector, and matrix operations 5 We write v R N as an expansion in coordinate basis vectors as v = v e [] + e [2] + +v N e [N] (2) where the components of e [ j] are Kroenecker deltas δ jk, e [ j] e [ j] e [ j] δ j 2 δ j2 = = δ jk = δ jn e [ j] N {, if j = k, if j k Addition of two real vectors v R N, w R N is straightforward, v w v + w v + w = + w 2 = + w 2 v N w N v N + w N as is multiplication of a vector v R N by a real scalar c R, v cv cv = c = c v N cv N (2) (22) (23) For all u, v, w R N and all c, c 2 R, u + (v + w) = (u + v) + w c(v + u) = cv + cu u + v = v + u (c + c 2 )v = c v + c 2 v (24) v + = v (c c 2 )v = c (c 2 v) v + ( v) = v = v where the null vector R N is = (25) We further add to the list of operations associated with the vectors v, w R N the dot (inner, scalar) product, v w = v w + w 2 + +v N w N = v k w k (26) k= wwwcambridgeorg
6 6 Linear algebra For example, for the two vectors v = 2 4 w = 5 (27) 3 6 v w = v w + w 2 + v 3 w 3 = ()(4) + (2)(5) + (3)(6) = = 32 (28) For 3-D vectors, the dot product is proportional to the product of the lengths and the cosine of the angle between the two vectors, where the length of v is v w = v w cos θ (29) v = v v (3) Therefore, when two vectors are parallel, the magnitude of their dot product is maximal and equals the product of their lengths, and when two vectors are perpendicular, their dot product is zero These ideas carry completely into N- dimensions The length of a vector v R N is v = v v = N vk 2 (3) If v w =, v and w are said to be orthogonal, the extension of the adjective perpendicular from R 3 to R N If v w = and v = w =, ie, both vectors are normalized to unit length, v and w are said to be orthonormal The formula for the length v of a vector v R N satisfies the more general properties of a norm v of v R N A norm v is a rule that assigns a real scalar, v R, to each vector v R N such that for every v, w R N, and for every c R,wehave k= v = v = if and only if (iff) v = cv = c v v + w v + w (32) Each norm also provides an accompanying metric, a measure of how different two vectors are d(v, w) = v w (33) In addition to the length, many other possible definitions of norm exist The p-norm, v p, of v R N is [ ] /p v p = v k p (34) k= wwwcambridgeorg
7 Review of scalar, vector, and matrix operations 7 Table p-norm values for the 3-D vector (, 2, 3) p v p = The length of a vector is thus also the 2-normFor v = [ 2 3], the values of the p-norm, computed from (35), are presented in Table v p = [ p + 2 p + 3 p ] /p = [() p + (2) p + (3) p ] /p (35) We define the infinity norm as the limit of v p as p, which merely extracts from v the largest magnitude of any component, v lim p v p = max j [,N] { v j } (36) For v = [ 2 3], v = 3 Like scalars, vectors can be complex We define the set of complex N-dimensional vectors as C N, and write each component of v C N as v j = a j + ib j a j, b j R i = (37) The complex conjugate of v C N, written as v or v *,is a + ib a ib v a 2 + ib 2 a 2 ib 2 = = a N + ib N a N ib N (38) For complex vectors v, w C N, to form the dot product v w, we take the complex conjugates of the first vector s components, v w = vk w k (39) This ensures that the length of any v C is always real and nonnegative, v 2 2 = vk v k = k= k= (a k ib k )(a k + ib k ) = k= For v, w C N, the order of the arguments is significant, k= ( a 2 k + bk 2 ) (4) v w = (w v) w v (4) wwwcambridgeorg
8 8 Linear algebra Matrix dimension For a linear system Ax = b, a a 2 a N a 2 a 22 a 2N A = a N a N2 a NN x x 2 x = x N b b 2 b = b N (42) to have a unique solution, there must be as many equations as unknowns, and so typically A will have an equal number N of columns and rows and thus be a square matrix A matrix is said to be of dimension M N if it has M rows and N columns We now consider some simple matrix operations Multiplication of an M N matrix A by a scalar c a a 2 a N ca ca 2 ca N a 2 a 22 a 2N ca = c = ca 2 ca 22 ca 2N a M a M2 a MN ca M ca M2 ca MN (43) Addition of an M N matrix A with an equal-sized M N matrix B a a N b b N a 2 a 2N + b 2 b 2N a M a MN b M b MN a + b a N + b N a 2 + b 2 a 2N + b 2N = a M + b M a MN + b MN (44) Note that A + B = B + A and that two matrices can be added only if both the number of rows and the number of columns are equal for each matrix Also, c(a + B) = ca+ cb Multiplication of a square N N matrix A with an N-dimensional vector v This operation must be defined as follows if we are to have equivalence between the coefficient and matrix/vector representations of a linear system: a a 2 a N v a v + a 2 + +a N v N a 2 a 22 a 2N Av = = a 2 v + a a 2N v N a N a N2 a NN v N a N v + a N2 + +a NN v N (45) wwwcambridgeorg
9 Review of scalar, vector, and matrix operations 9 Av is also an N-dimensional vector, whose j th component is (Av) j = a j v + a j2 + +a jn v N = a jk v k (46) k= We compute (Av) j by summing a jk v k along rows of A and down the vector, [ a jk ] v k Multiplication of an M N matrix A with an N-dimensional vector v From the rule for forming Av, we see that the number of columns of A must equal the dimension of v; however, we also can define Av when M N, a a 2 a N v a v + a 2 + +a N v N a 2 a 22 a 2N Av = = a 2 v + a a 2N v N a M a M2 a MN v N a M v + a M2 + +a MN v N (47) If v R N, for an M N matrix A, Av R M Consider the following examples: = = 32 (48) Note also that A(cv) = cav and A(v + w) = Av + Aw Matrix transposition We define for an M N matrix A the transpose A T to be the N M matrix a a 2 a N A T a 2 a 22 a 2N = a M a M2 a MN T a a 2 a M a 2 a 22 a M2 = a N a 2N a NM (49) wwwcambridgeorg
10 Linear algebra The transpose operation is essentially a mirror reflection across the principal diagonal a, a 22, a 33, Consider the following examples: T [ ] T = = (5) If a matrix is equal to its transpose, A = A T, it is said to be symmetric Then, a ij = ( A T) ij = a ji i, j {, 2,, N} (5) Complex-valued matrices Here we have defined operations for real matrices; however, matrices may also be complexvalued, c c N (a + ib 2 ) (a N + ib N ) c 2 c 2N C = = (a 2 + ib 2 ) (a 2N + ib 2N ) (52) c M c MN (a M + ib M ) (a MN + ib MN ) For the moment, we are concerned with the properties of real matrices, as applied to solving linear systems in which the coefficients are real Vectors as matrices Finally, we note that the matrix operations above can be extended to vectors by considering a vector v R N to be an N matrix if in column form and to be a N matrix if in row form Thus, for v, w R N, expressing vectors by default as column vectors, we write the dot product as v w = v T w = [v v N ] w w N = v w + +v N w N (53) The notation v T w for the dot product v w is used extensively in this text Elimination methods for solving linear systems With these basic definitions in hand, we now begin to consider the solution of the linear system Ax = b, in which x, b R N and A is an N N real matrix We consider here elimination methods in which we convert the linear system into an equivalent one that is easier to solve These methods are straightforward to implement and work generally for any linear system that has a unique solution; however, they can be quite costly (perhaps prohibitively so) for large systems Later, we consider iterative methods that are more effective for certain classes of large systems wwwcambridgeorg
Matrix Operations. Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix
Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix Matrix Operations Matrix Addition and Matrix Scalar Multiply Matrix Multiply Matrix
More informationMobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti
Mobile Robotics 1 A Compact Course on Linear Algebra Giorgio Grisetti SA-1 Vectors Arrays of numbers They represent a point in a n dimensional space 2 Vectors: Scalar Product Scalar-Vector Product Changes
More informationChapter 2. Linear Algebra. rather simple and learning them will eventually allow us to explain the strange results of
Chapter 2 Linear Algebra In this chapter, we study the formal structure that provides the background for quantum mechanics. The basic ideas of the mathematical machinery, linear algebra, are rather simple
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
More informationIntroduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Bastian Steder
Introduction to Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Bastian Steder Reference Book Thrun, Burgard, and Fox: Probabilistic Robotics Vectors Arrays of numbers Vectors represent
More informationThis appendix provides a very basic introduction to linear algebra concepts.
APPENDIX Basic Linear Algebra Concepts This appendix provides a very basic introduction to linear algebra concepts. Some of these concepts are intentionally presented here in a somewhat simplified (not
More informationLecture 3: QR-Factorization
Lecture 3: QR-Factorization This lecture introduces the Gram Schmidt orthonormalization process and the associated QR-factorization of matrices It also outlines some applications of this factorization
More informationMathematical foundations - linear algebra
Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar
More informationB553 Lecture 5: Matrix Algebra Review
B553 Lecture 5: Matrix Algebra Review Kris Hauser January 19, 2012 We have seen in prior lectures how vectors represent points in R n and gradients of functions. Matrices represent linear transformations
More informationIntroduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello
Introduction to Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello Vectors Arrays of numbers Vectors represent a point
More informationMatrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A =
30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A = a 11 a 12... a 1N a 21 a 22... a 2N...... a M1 a M2... a MN A matrix can
More informationMATRICES. a m,1 a m,n A =
MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationLinear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations.
POLI 7 - Mathematical and Statistical Foundations Prof S Saiegh Fall Lecture Notes - Class 4 October 4, Linear Algebra The analysis of many models in the social sciences reduces to the study of systems
More information1 Matrices and vector spaces
Matrices and vector spaces. Which of the following statements about linear vector spaces are true? Where a statement is false, give a counter-example to demonstrate this. (a) Non-singular N N matrices
More informationAM 205: lecture 8. Last time: Cholesky factorization, QR factorization Today: how to compute the QR factorization, the Singular Value Decomposition
AM 205: lecture 8 Last time: Cholesky factorization, QR factorization Today: how to compute the QR factorization, the Singular Value Decomposition QR Factorization A matrix A R m n, m n, can be factorized
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationA = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
More informationAPPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the n-dimensional column vector with components x 1 x 2.
APPENDIX A Background Mathematics A. Linear Algebra A.. Vector algebra Let x denote the n-dimensional column vector with components 0 x x 2 B C @. A x n Definition 6 (scalar product). The scalar product
More informationIntroduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Introduction to Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional space
More informationReview of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem
Review of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem Steven J. Miller June 19, 2004 Abstract Matrices can be thought of as rectangular (often square) arrays of numbers, or as
More informationA VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationEE731 Lecture Notes: Matrix Computations for Signal Processing
EE731 Lecture Notes: Matrix Computations for Signal Processing James P. Reilly c Department of Electrical and Computer Engineering McMaster University September 22, 2005 0 Preface This collection of ten
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationFormula for the inverse matrix. Cramer s rule. Review: 3 3 determinants can be computed expanding by any row or column
Math 20F Linear Algebra Lecture 18 1 Determinants, n n Review: The 3 3 case Slide 1 Determinants n n (Expansions by rows and columns Relation with Gauss elimination matrices: Properties) Formula for the
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p
More informationACM106a - Homework 2 Solutions
ACM06a - Homework 2 Solutions prepared by Svitlana Vyetrenko October 7, 2006. Chapter 2, problem 2.2 (solution adapted from Golub, Van Loan, pp.52-54): For the proof we will use the fact that if A C m
More informationBasic Concepts in Linear Algebra
Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More informationMath 3191 Applied Linear Algebra
Math 191 Applied Linear Algebra Lecture 1: Inner Products, Length, Orthogonality Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/ Motivation Not all linear systems have
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)
AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) Lecture 1: Course Overview; Matrix Multiplication Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical
More informationReview of linear algebra
Review of linear algebra 1 Vectors and matrices We will just touch very briefly on certain aspects of linear algebra, most of which should be familiar. Recall that we deal with vectors, i.e. elements of
More informationReview of Basic Concepts in Linear Algebra
Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra
More informationLinear algebra. 1.1 Numbers. d n x = 10 n (1.1) n=m x
1 Linear algebra 1.1 Numbers The natural numbers are the positive integers and zero. Rational numbers are ratios of integers. Irrational numbers have decimal digits d n d n x = 10 n (1.1 n=m x that do
More informationElementary Linear Algebra
Matrices J MUSCAT Elementary Linear Algebra Matrices Definition Dr J Muscat 2002 A matrix is a rectangular array of numbers, arranged in rows and columns a a 2 a 3 a n a 2 a 22 a 23 a 2n A = a m a mn We
More information1 Inner Product and Orthogonality
CSCI 4/Fall 6/Vora/GWU/Orthogonality and Norms Inner Product and Orthogonality Definition : The inner product of two vectors x and y, x x x =.., y =. x n y y... y n is denoted x, y : Note that n x, y =
More informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationThe Singular Value Decomposition
The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will
More informationMatrices. Chapter What is a Matrix? We review the basic matrix operations. An array of numbers a a 1n A = a m1...
Chapter Matrices We review the basic matrix operations What is a Matrix? An array of numbers a a n A = a m a mn with m rows and n columns is a m n matrix Element a ij in located in position (i, j The elements
More informationMatrices and systems of linear equations
Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 1: Course Overview & Matrix-Vector Multiplication Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 20 Outline 1 Course
More informationThe Solution of Linear Systems AX = B
Chapter 2 The Solution of Linear Systems AX = B 21 Upper-triangular Linear Systems We will now develop the back-substitution algorithm, which is useful for solving a linear system of equations that has
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Revised February, 1982 NOTES ON MATRIX METHODS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Revised February, 198 NOTES ON MATRIX METHODS 1. Matrix Algebra Margenau and Murphy, The Mathematics of Physics and Chemistry, Chapter 10, give almost
More informationMidterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015
Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015 The test lasts 1 hour and 15 minutes. No documents are allowed. The use of a calculator, cell phone or other equivalent electronic
More informationImage Registration Lecture 2: Vectors and Matrices
Image Registration Lecture 2: Vectors and Matrices Prof. Charlene Tsai Lecture Overview Vectors Matrices Basics Orthogonal matrices Singular Value Decomposition (SVD) 2 1 Preliminary Comments Some of this
More informationVectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1. x 2. x =
Linear Algebra Review Vectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1 x x = 2. x n Vectors of up to three dimensions are easy to diagram.
More information33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM
33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM (UPDATED MARCH 17, 2018) The final exam will be cumulative, with a bit more weight on more recent material. This outline covers the what we ve done since the
More informationMathematical Preliminaries and Review
Mathematical Preliminaries and Review Ruye Wang for E59 1 Complex Variables and Sinusoids AcomplexnumbercanberepresentedineitherCartesianorpolarcoordinatesystem: z = x + jy= z z = re jθ = r(cos θ + j sin
More informationAssignment 10. Arfken Show that Stirling s formula is an asymptotic expansion. The remainder term is. B 2n 2n(2n 1) x1 2n.
Assignment Arfken 5.. Show that Stirling s formula is an asymptotic expansion. The remainder term is R N (x nn+ for some N. The condition for an asymptotic series, lim x xn R N lim x nn+ B n n(n x n B
More informationLinear Algebra V = T = ( 4 3 ).
Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional
More informationSection 9.2: Matrices.. a m1 a m2 a mn
Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in
More informationPage 52. Lecture 3: Inner Product Spaces Dual Spaces, Dirac Notation, and Adjoints Date Revised: 2008/10/03 Date Given: 2008/10/03
Page 5 Lecture : Inner Product Spaces Dual Spaces, Dirac Notation, and Adjoints Date Revised: 008/10/0 Date Given: 008/10/0 Inner Product Spaces: Definitions Section. Mathematical Preliminaries: Inner
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J Olver 3 Review of Matrix Algebra Vectors and matrices are essential for modern analysis of systems of equations algebrai, differential, functional, etc In this
More informationThe Transpose of a Vector
8 CHAPTER Vectors The Transpose of a Vector We now consider the transpose of a vector in R n, which is a row vector. For a vector u 1 u. u n the transpose is denoted by u T = [ u 1 u u n ] EXAMPLE -5 Find
More informationAM 205: lecture 6. Last time: finished the data fitting topic Today s lecture: numerical linear algebra, LU factorization
AM 205: lecture 6 Last time: finished the data fitting topic Today s lecture: numerical linear algebra, LU factorization Unit II: Numerical Linear Algebra Motivation Almost everything in Scientific Computing
More information10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )
c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side
More informationCLASS 12 ALGEBRA OF MATRICES
CLASS 12 ALGEBRA OF MATRICES Deepak Sir 9811291604 SHRI SAI MASTERS TUITION CENTER CLASS 12 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationIntroduction - Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim
Introduction - Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its
More informationVectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes.
Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes Matrices and linear equations A matrix is an m-by-n array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a
More informationA Review of Matrix Analysis
Matrix Notation Part Matrix Operations Matrices are simply rectangular arrays of quantities Each quantity in the array is called an element of the matrix and an element can be either a numerical value
More informationRepeated Eigenvalues and Symmetric Matrices
Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one
More informationb 1 b 2.. b = b m A = [a 1,a 2,...,a n ] where a 1,j a 2,j a j = a m,j Let A R m n and x 1 x 2 x = x n
Lectures -2: Linear Algebra Background Almost all linear and nonlinear problems in scientific computation require the use of linear algebra These lectures review basic concepts in a way that has proven
More information3 a 21 a a 2N. 3 a 21 a a 2M
APPENDIX: MATHEMATICS REVIEW G 12.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers 2 A = 6 4 a 11 a 12... a 1N a 21 a 22... a 2N. 7..... 5 a M1 a M2...
More informationApplied Mathematics 205. Unit II: Numerical Linear Algebra. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit II: Numerical Linear Algebra Lecturer: Dr. David Knezevic Unit II: Numerical Linear Algebra Chapter II.3: QR Factorization, SVD 2 / 66 QR Factorization 3 / 66 QR Factorization
More informationMATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra
MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra A. Vectors A vector is a quantity that has both magnitude and direction, like velocity. The location of a vector is irrelevant;
More informationChapter 5 Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n
More informationReduction to the associated homogeneous system via a particular solution
June PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 5) Linear Algebra This study guide describes briefly the course materials to be covered in MA 5. In order to be qualified for the credit, one
More informationMatrix Algebra: Vectors
A Matrix Algebra: Vectors A Appendix A: MATRIX ALGEBRA: VECTORS A 2 A MOTIVATION Matrix notation was invented primarily to express linear algebra relations in compact form Compactness enhances visualization
More informationAlgebra C Numerical Linear Algebra Sample Exam Problems
Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric
More information22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes
Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one
More informationProfessor Terje Haukaas University of British Columbia, Vancouver Notation
Notation This document establishes the notation that is employed throughout these notes. It is intended as a look-up source during the study of other documents and software on this website. As a general
More informationTopics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij
Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix
More informationELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices
ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex
More informationMath 2331 Linear Algebra
6.1 Inner Product, Length & Orthogonality Math 2331 Linear Algebra 6.1 Inner Product, Length & Orthogonality Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/
More informationA Review of Linear Algebra
A Review of Linear Algebra Gerald Recktenwald Portland State University Mechanical Engineering Department gerry@me.pdx.edu These slides are a supplement to the book Numerical Methods with Matlab: Implementations
More informationLinear Algebra Primer
Introduction Linear Algebra Primer Daniel S. Stutts, Ph.D. Original Edition: 2/99 Current Edition: 4//4 This primer was written to provide a brief overview of the main concepts and methods in elementary
More informationPhys 201. Matrices and Determinants
Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1
More informationv = v 1 2 +v 2 2. Two successive applications of this idea give the length of the vector v R 3 :
Length, Angle and the Inner Product The length (or norm) of a vector v R 2 (viewed as connecting the origin to a point (v 1,v 2 )) is easily determined by the Pythagorean Theorem and is denoted v : v =
More informationPolar Form of a Complex Number (I)
Polar Form of a Complex Number (I) The polar form of a complex number z = x + iy is given by z = r (cos θ + i sin θ) where θ is the angle from the real axes to the vector z and r is the modulus of z, i.e.
More informationj=1 u 1jv 1j. 1/ 2 Lemma 1. An orthogonal set of vectors must be linearly independent.
Lecture Notes: Orthogonal and Symmetric Matrices Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Orthogonal Matrix Definition. Let u = [u
More informationMatrix & Linear Algebra
Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A
More informationA TOUR OF LINEAR ALGEBRA FOR JDEP 384H
A TOUR OF LINEAR ALGEBRA FOR JDEP 384H Contents Solving Systems 1 Matrix Arithmetic 3 The Basic Rules of Matrix Arithmetic 4 Norms and Dot Products 5 Norms 5 Dot Products 6 Linear Programming 7 Eigenvectors
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationLinear Equations and Matrix
1/60 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear
More informationBasic Concepts in Matrix Algebra
Basic Concepts in Matrix Algebra An column array of p elements is called a vector of dimension p and is written as x p 1 = x 1 x 2. x p. The transpose of the column vector x p 1 is row vector x = [x 1
More information22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices
m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationCourse 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations
Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.
More informationLecture 8 : Eigenvalues and Eigenvectors
CPS290: Algorithmic Foundations of Data Science February 24, 2017 Lecture 8 : Eigenvalues and Eigenvectors Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Hermitian Matrices It is simpler to begin with
More informationNumerical Linear Algebra
Numerical Linear Algebra The two principal problems in linear algebra are: Linear system Given an n n matrix A and an n-vector b, determine x IR n such that A x = b Eigenvalue problem Given an n n matrix
More informationMathematics. EC / EE / IN / ME / CE. for
Mathematics for EC / EE / IN / ME / CE By www.thegateacademy.com Syllabus Syllabus for Mathematics Linear Algebra: Matrix Algebra, Systems of Linear Equations, Eigenvalues and Eigenvectors. Probability
More informationGetting Started with Communications Engineering. Rows first, columns second. Remember that. R then C. 1
1 Rows first, columns second. Remember that. R then C. 1 A matrix is a set of real or complex numbers arranged in a rectangular array. They can be any size and shape (provided they are rectangular). A
More informationLinear Algebra (Review) Volker Tresp 2018
Linear Algebra (Review) Volker Tresp 2018 1 Vectors k, M, N are scalars A one-dimensional array c is a column vector. Thus in two dimensions, ( ) c1 c = c 2 c i is the i-th component of c c T = (c 1, c
More information