MATRIX ALGEBRA. or x = (x 1,..., x n ) R n. y 1 y 2. x 2. x m. y m. y = cos θ 1 = x 1 L x. sin θ 1 = x 2. cos θ 2 = y 1 L y.
|
|
- Elaine Hall
- 6 years ago
- Views:
Transcription
1 as Basics Vectors MATRIX ALGEBRA An array of n real numbers x, x,, x n is called a vector and it is written x = x x n or x = x,, x n R n prime operation=transposing a column to a row Basic vector operations Multiplication with a constant c cx { cx cx = if c < c > cx m contraction expantion Addition of x, y R n x + y = x x + y y = x + y x + y x m y m x m + y m Inner Product or Scalar Product x = x y = y x n y n x y = x y + x y + + x n y n L x = length of x = x x Let x, y R and θ i denotes the angle between the vector and the x axis Then cos θ = x L x sin θ = x L x cos θ = y L y sin θ = y L y θ = angle between x and y θ = θ θ cos θ = cosθ θ = cos θ cos θ + sin θ sin θ
2 In general for x, y R n cos θ = x L x y L y + x L x y L y = x y L x Ly cos θ = x y L x L y or x y = L x L y cos θ Remark: cos θ = x and y are perpendicular Length of a vector x R n : L x = x + x + + x n How multiplication with a constant c changes the length? L x = c x + c x + + c x n = c L x Remarks: Let c = L x, then is a vector with unit length and with direction of x L x x If ax + bx = then x = b a x and x have the same direction but different length Unit vectors in R : e = e = Unit vectors in R n : Let e = x = x e = e n = = x e + x e + + x m e n x n Definition: The space of all n-tuples with scalar multiplication and addition as defined above, is called a vector space Definition: by = a x + a x + + a k x k is a linear combination of the vectors x,, x k The zero vector is defined as =
3 Definition: The vectors x, x,, x k are said to be linearly dependent if there exist k numbers a,, a k not all zero, such that a x + + a k x k = Otherwise x, x,, x k is said to be linearly independent Examples: i e = and e = are linearly independent, because if a e + a e = then a a = a = a = ii Similarly you can prove in R n that e,, e n are linearly independent iii Let x = x = 5 x 3 = Then x, x and x 3 are linearly dependent since x x + 3x 3 = Definition: A set of m linearly independent vectors in R m called a basis for the vector space of m-tuples Theorem: basis Every vector in R m can be expressed as a unique linear combination of a fixed Example Let e,, e m be a basis in R m Then x = x e + x e + + x m e m Definition: The length of a vector x is L x = x + x + + x m Definition: The inner product or dot product of two vectors x, y R m is x y = y x = x y + x y + + x m y m Remark: ilength of a vector x: L x = x x 3
4 ii Let us denote θ the angle between two vectors x, y R m Then cosθ = x y x x y y Definition: When the angle between two vectors x and y is θ = 9 or 7 we say that x and y are perpendicular or orthogonal Since cos 9 = cos 7 = x and y are perpendicular if x y = Notation Example x y The basis vectors e,, e m are mutually perpendicular e ie j =, i j bf Remark: For any vector x, the unit vector with direction x is fracx x x Results: a z is perpendicular to every vector x R m z = b If z is perpendicular to each vector x,, x k then z is perpendicular to their linear span If z x i =, i =,, k then z a x + a x + + a k x k = c Mutually perpendicular vectors are linearly independent Definition: The projection of x on y P y x = x y y L y If L y = then P y x = x yy Lemma: Let z = x y L y y Then x z is orthogonal to y Proof: y x x y y = y x x y y y L y L y = 4
5 GRAM-SCHMIDT ORTHOGONALIZATION PROCESS: Given linearly independent vectors x,, x k, there exists mutually perpendicular vectors u,, u k with the same linear span This may be constructed sequentially by setting u = x u = x x u u u u u 3 = x 3 x 3 u u u u x 3 u u u u u k = x k x k u u u u x k u k u u k u k k We can use unit length vectors z,, z k instead of u,, u k ; z j = u j u j u j j =,, k Example Compute u, u, z, z for x = 4 x = 3 5
6 MATRICES A matrix is a rectangular array of real numbers An m k columns A, R, Σ boldface letters, denote matrices a a a k a A m k = a a k a m a m a mk matrix has m rows and k A = {a ij } i =,, m, j =,, k For example an m matrix has m rows and column, it is an m dimensional vector or column matrix m matrix is a column vector, m matrix is a row vector Definition: Let A = a ij B = b ij be two m k matrices We say that the two matrices are equal A = B a ij = b ij i, j I e two matrices are equal if i their dimensionality is the same, ii every corresponding element is the same MATRIX OPERATIONS Let A, B be two m k matrices Matrix Addition: C = A + B is an m k matrix with elements c ij = a ij + b ij i =,, m j =,, k Scalar Multiplication: c arbitrary scalar A = a ij ca = Ac = B = b ij where b ij = ca ij = a ij c i =,, m j =,, k Matrix Substraction A = a ij B = b ij i =,, m j =,, k A B = A + B = C where C = c ij c ij = a ij b ij i =,, m, j =,, k Definition: Transpose of a matrix A = a ij, i =,, m j =,, k, A is defined as a k m matrix with element a ji j =,, k i =,, m That is the transpose of a matrix A is obtained from A by interchanging the rows and columns 6
7 Example: A = 3 A = 3 Theorem: Let A, B, C be m k matrices, c, d are scalars a A + B + C = A + B + C b A + B = B + A c ca + B = ca + cb d c + da = ca + da e A + B = A + B f cda = cda g ca = ca Definition:If numbers of rows are equal to the numbers columns of a matrix A, then it is called square matrix Definition: Let A be a k k square matrix The matrix A is said to be symmetric if A = A That is A is symmetric a ij = a ji i, j =,, k Examples: i Let I denotes the m m matrix with -s in the main diagonal and zeros elswhere Then I is symmetric ii 4 A = 4 Definition: I the k k matrix is defined as the identity matrix if it has ones only in the main diagonal and zeros elsewhere Definition: Matrix Product Suppose that A = a ij is an m n matrix B = b ij is an n k matrix Then AB = C = c ij, where C is and m k matrix with elements c ij, i =,, m j =,, k, where c ij is the scalar product of ith row of A with jth column of B, n c ij = a il b lj i =,, m j =,, k l= Example: A = 3 = 3 B = 3 = AB = = 4 4
8 Theorem: Properties of matrix multiplication A, B, C defined such that the indicated products are defined and a scalar c is given acab = cab babc = ABC cab + C = AB + AC db + CA = BA + CA eab = B A Important Remarks: AB BA AB = does not imply that A =, or B = Example: 3 4 = However, it is true, that if A = or B = then A B = Definition: The row rank of a matrix A is the maximum number of linearly independent rows, considered as vectors The column rank of A is the maximum number of independent columns considered as vectors Theorem: For any matrix A, the number of independent rows is equal to the number of independent columns Definition: Let A be a k k square matrix The matrix A is nonsingular if Ax = x = where x is a k-dimensional vector A is singular if there exists x such that Ax = Remark: Let a x + a x + + a k x k a Ax = x + a x + + a k x k a k x + a k x + + a kk x k = x a + x a + + x k a k where a,, a k are the column vectors of A Thus, for a nonsingular matrix A x a + + x k a k = x = x = = x k = i e nonsingularty is equivalent that the columns of A are linearly independent 8
9 Theorem: Let A be a k k nonsingular matrix Then there exists one and only one k k matrix B such that AB = BA = I where I is the identity Definition: The matrix B, such that AB = BA = I is called the inverse of A and it is denoted by A In fact if BA = I or AB = I then B = A 4 3 Example: A = A 3 = AA = = = I Theorem: a Inverse of a matrix A = a a a a A = A is given by a a a a b In general A is given the following way a ij = A A ji i+j where A ij is the value of the determinant of A ji obtained from A deleting the j-th row and i-th column of A Definition: The determinant of a square matrix A is a scalar denoted by A defined recursively if k = A = a, k A = a ij A ij i+j j= where A ij is the determinant of the k k matrix, obtained as deleting the ith row and jth column of A We can use the first row: A = k a j A j +j j= Theorem: The following statements are equivalent for a k k square matrix A; a Ax = x = ie A nonsingular 9
10 b A c there exists a matrix A such that AA = A A = I Theorem: Let A and B be k k square matrices, and suppose that both have inverse matrix Then a A = A b AB = B A Proof: a I = A A = A A b ABAB = ABB A = I Theorem: Let A and B be k k matrices a A = A b If each element of a row column of A is zero then A = c If any two rows or columns of A are identical, then A = d If A is nonsingular, then A = ie A A = A e AB = A B f ca = c k A Definition: Let A = a ij be a k k square matrix The trace of the matrix A is tra = k a ii i= Theorem: Let A, B be k k matrices, c scalar Then a trca = ctra b tra ± B = tra ± trb ctrab = trba d trb AB = tra e traa = k k a ij i= j= Definition: A square matrix A is said to be orthogonal if its rows considered as vectors are mutually perpendicular and have unit length, which means that AA = I Theorem: A matrix A is orthogonal if and only if A = A For an orthogonal matrix AA = A A = I Example: A = A A = is orthogonal A = = = I
11 Definiton: Let A be a k k matrix and I be the k k identity Then the scalars λ, λ,, λ k satisfy the polynomial equation A λi = called eigenvalues of A A λi = called characteristic equation λ Example: A = A λi = λ A λi = λ4 λ = λ = λ = 4 are the eigenvalues Definiton: Let A be a k k matrix Let λ be an eigenvalue of A If x such that Ax = λx then x is said to be an eigenvector of A associated with the eigenvalue λ Ax = λx = A λix = x col A λi + x col A λi + + x k col k A λi If x it means that the columns of A λi are linearly dependent Example: A = λ = λ = From the first expression, x x x x = = x x 4x 4x x = x 3x + 4x = x x = x = From the second expression x = 4x 3x + 4x = 4x x = x = Usual practise to determine an eigenvector with length one In the example: e i = x i x i x i i =, e = x e = x
12 Definition: Quadratic form Qx in k variables x,, x k, where x = x,, x k R k, is defined as Qx = x Ax where A is a fix k k matrix The quadratic form is a quadratic function of x, x,, x k Example: x, x x x = x, x x + x x x x + x x x = x + x x + x = x + x Theorem Let A be a k k symmetric matrix, i e A = A then A has k pairs of eigenvalues and eigenvectors λ, e, λ, e,, λ k, e k The eigenvectors can be chosen to satisfy = e e = = e ke k and be mutually perpendicular The eigenvectors are unique unless two or more eigenvalues are equal Definition: Positive Definite Matrices Let A be a symmetric matrix k k A is said to be positive definite if x Ax > for all x R k x A is positive semi-definite if x Ax for all x R k Theorem: Spectral decomposition of a symmetric k k matrix A is given: A = λ e e + λ e e + + λ k e k e k λ,, λ k are eigenvalues of A and e,, e k are the associated normalized eigenvectors Theorem A, k k symmetric matrix is positive definite if and only if every eigenvalue of A is positive A is positive semi-definite if and only if every eigenvalue of A is nonnegative Proof: Trivial from the spectral decomposition theorem A = λ e e + λ e e + + λ k e k e k x Ax = λ x e e x + λ x e e x + + λ k x ke k e kx = λ y + λ y + + λ k y k where y i = x e i Choosing x = e j j =,, k follows the theorem
13 Another form of the spectral decomposition: λ A = e e k e = PΛP λ k e k INVERZE AND SQUARE ROOT OF A POSITIVE DEFINIT MATRIX By the spectral decomposition, if A is positive definit: A = PΛP Inverse of A If A is positive definite, then the eigenvaluse of A are λ λ, λ λ k > i =,, k The inverse of A is A = PΛ P where Λ = Because; A A = PΛ P PΛP = PΛ ΛP = PP = I Square Root Matrix defined for positive semi-definite A Notice: A A = PΛ P PΛ P = A A = A k A = λ i e i e i = PΛ P i= λ k MATRIX INEQUALITIES AND MAXIMIZATION Cauchy-Schwarz Inequality: Let b, d R p, then b d b bd d, with equality if and only if b = cd Proof: The inequality is obvious when one of the vectors is the zero one For b xd we have that, < b xd b xd = b b xb d xd b + x d d = b b xb d + x d d Adding and substratcting b d /d d, we have that < b b b d d d + d d x b d d d 3
14 Choosing x = b d/d d follows the statement Extended Cauchy Schwartz Inequality: Let b, d R p and let B be a positive definite p p matrix Then b d b Bbd B d, with equality if and only if b = cb d for some constant c Maximization of Quadratic forms on the Unit Sphere: Let B be a p p positive definite matrix with eigenvectors e,, e p and associated eigenvalues λ,, λ p eigenvectors, where λ λ λ p Then Morover max x x Bx x x = λ x Bx min x x x x Bx max x e,,e k x x and the equality is attained when x = e k+ = λ k+, = λ p 4
2. Matrix Algebra and Random Vectors
2. Matrix Algebra and Random Vectors 2.1 Introduction Multivariate data can be conveniently display as array of numbers. In general, a rectangular array of numbers with, for instance, n rows and p columns
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 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 informationMath Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p
Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the
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 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 informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationStat 206: Linear algebra
Stat 206: Linear algebra James Johndrow (adapted from Iain Johnstone s notes) 2016-11-02 Vectors We have already been working with vectors, but let s review a few more concepts. The inner product of two
More informationMore Linear Algebra. Edps/Soc 584, Psych 594. Carolyn J. Anderson
More Linear Algebra Edps/Soc 584, Psych 594 Carolyn J. Anderson Department of Educational Psychology I L L I N O I S university of illinois at urbana-champaign c Board of Trustees, University of Illinois
More informationLinear Algebra Highlights
Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to
More informationMassachusetts Institute of Technology Department of Economics Statistics. Lecture Notes on Matrix Algebra
Massachusetts Institute of Technology Department of Economics 14.381 Statistics Guido Kuersteiner Lecture Notes on Matrix Algebra These lecture notes summarize some basic results on matrix algebra used
More informationKnowledge Discovery and Data Mining 1 (VO) ( )
Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory
More informationAppendix A: Matrices
Appendix A: Matrices A matrix is a rectangular array of numbers Such arrays have rows and columns The numbers of rows and columns are referred to as the dimensions of a matrix A matrix with, say, 5 rows
More informationChapter 1. Matrix Algebra
ST4233, Linear Models, Semester 1 2008-2009 Chapter 1. Matrix Algebra 1 Matrix and vector notation Definition 1.1 A matrix is a rectangular or square array of numbers of variables. We use uppercase boldface
More informationIntroduc)on to linear algebra
Introduc)on to linear algebra Vector A vector, v, of dimension n is an n 1 rectangular array of elements v 1 v v = 2 " v n % vectors will be column vectors. They may also be row vectors, when transposed
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 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 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 informationExercise Sheet 1.
Exercise Sheet 1 You can download my lecture and exercise sheets at the address http://sami.hust.edu.vn/giang-vien/?name=huynt 1) Let A, B be sets. What does the statement "A is not a subset of B " mean?
More informationMatrix Algebra, part 2
Matrix Algebra, part 2 Ming-Ching Luoh 2005.9.12 1 / 38 Diagonalization and Spectral Decomposition of a Matrix Optimization 2 / 38 Diagonalization and Spectral Decomposition of a Matrix Also called Eigenvalues
More informationIntroduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX
Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX September 2007 MSc Sep Intro QT 1 Who are these course for? The September
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 informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationQuick Tour of Linear Algebra and Graph Theory
Quick Tour of Linear Algebra and Graph Theory CS224W: Social and Information Network Analysis Fall 2014 David Hallac Based on Peter Lofgren, Yu Wayne Wu, and Borja Pelato s previous versions Matrices and
More informationVectors and Matrices Statistics with Vectors and Matrices
Vectors and Matrices Statistics with Vectors and Matrices Lecture 3 September 7, 005 Analysis Lecture #3-9/7/005 Slide 1 of 55 Today s Lecture Vectors and Matrices (Supplement A - augmented with SAS proc
More informationDef. The euclidian distance between two points x = (x 1,...,x p ) t and y = (y 1,...,y p ) t in the p-dimensional space R p is defined as
MAHALANOBIS DISTANCE Def. The euclidian distance between two points x = (x 1,...,x p ) t and y = (y 1,...,y p ) t in the p-dimensional space R p is defined as d E (x, y) = (x 1 y 1 ) 2 + +(x p y p ) 2
More informationA Little Necessary Matrix Algebra for Doctoral Studies in Business & Economics. Matrix Algebra
A Little Necessary Matrix Algebra for Doctoral Studies in Business & Economics James J. Cochran Department of Marketing & Analysis Louisiana Tech University Jcochran@cab.latech.edu Matrix Algebra Matrix
More informationSTAT200C: Review of Linear Algebra
Stat200C Instructor: Zhaoxia Yu STAT200C: Review of Linear Algebra 1 Review of Linear Algebra 1.1 Vector Spaces, Rank, Trace, and Linear Equations 1.1.1 Rank and Vector Spaces Definition A vector whose
More informationMathematical Foundations of Applied Statistics: Matrix Algebra
Mathematical Foundations of Applied Statistics: Matrix Algebra Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/105 Literature Seber, G.
More informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationMath Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT
Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
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 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 informationLinear Models Review
Linear Models Review Vectors in IR n will be written as ordered n-tuples which are understood to be column vectors, or n 1 matrices. A vector variable will be indicted with bold face, and the prime sign
More informationSome notes on Linear Algebra. Mark Schmidt September 10, 2009
Some notes on Linear Algebra Mark Schmidt September 10, 2009 References Linear Algebra and Its Applications. Strang, 1988. Practical Optimization. Gill, Murray, Wright, 1982. Matrix Computations. Golub
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
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 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 informationLinear Algebra: Characteristic Value Problem
Linear Algebra: Characteristic Value Problem . The Characteristic Value Problem Let < be the set of real numbers and { be the set of complex numbers. Given an n n real matrix A; does there exist a number
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
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 informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationMatrix Algebra: Summary
May, 27 Appendix E Matrix Algebra: Summary ontents E. Vectors and Matrtices.......................... 2 E.. Notation.................................. 2 E..2 Special Types of Vectors.........................
More information3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions
A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence
More informationLecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε,
2. REVIEW OF LINEAR ALGEBRA 1 Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε, where Y n 1 response vector and X n p is the model matrix (or design matrix ) with one row for
More informationMath Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More informationEcon Slides from Lecture 7
Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
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 informationReview of Vectors and Matrices
A P P E N D I X D Review of Vectors and Matrices D. VECTORS D.. Definition of a Vector Let p, p, Á, p n be any n real numbers and P an ordered set of these real numbers that is, P = p, p, Á, p n Then P
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 informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
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 informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization
ICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 21 Symmetric matrices An n n
More informationLinear Algebra. Session 12
Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)
More informationNotes on Linear Algebra and Matrix Theory
Massimo Franceschet featuring Enrico Bozzo Scalar product The scalar product (a.k.a. dot product or inner product) of two real vectors x = (x 1,..., x n ) and y = (y 1,..., y n ) is not a vector but a
More informationMatrices and Linear Algebra
Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationProblem Set 1. Homeworks will graded based on content and clarity. Please show your work clearly for full credit.
CSE 151: Introduction to Machine Learning Winter 2017 Problem Set 1 Instructor: Kamalika Chaudhuri Due on: Jan 28 Instructions This is a 40 point homework Homeworks will graded based on content and clarity
More informationProblem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show
MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,
More informationMATH 106 LINEAR ALGEBRA LECTURE NOTES
MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL - These Lecture Notes are not in a final form being still subject of improvement Contents Systems of linear equations and matrices 5 Introduction to systems of
More informationMatrices A brief introduction
Matrices A brief introduction Basilio Bona DAUIN Politecnico di Torino Semester 1, 2014-15 B. Bona (DAUIN) Matrices Semester 1, 2014-15 1 / 41 Definitions Definition A matrix is a set of N real or complex
More informationIntroduction to Matrices
POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder
More informationLinear algebra for computational statistics
University of Seoul May 3, 2018 Vector and Matrix Notation Denote 2-dimensional data array (n p matrix) by X. Denote the element in the ith row and the jth column of X by x ij or (X) ij. Denote by X j
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationLinear Algebra and Matrices
Linear Algebra and Matrices 4 Overview In this chapter we studying true matrix operations, not element operations as was done in earlier chapters. Working with MAT- LAB functions should now be fairly routine.
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationQuick Tour of Linear Algebra and Graph Theory
Quick Tour of Linear Algebra and Graph Theory CS224w: Social and Information Network Analysis Fall 2012 Yu Wayne Wu Based on Borja Pelato s version in Fall 2011 Matrices and Vectors Matrix: A rectangular
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More informationMAT 610: Numerical Linear Algebra. James V. Lambers
MAT 610: Numerical Linear Algebra James V Lambers January 16, 2017 2 Contents 1 Matrix Multiplication Problems 7 11 Introduction 7 111 Systems of Linear Equations 7 112 The Eigenvalue Problem 8 12 Basic
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationChapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More information1 Matrices and matrix algebra
1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. For instance 4 2 0 3 1 A = 5 1.2 0.7 x 3 π 3 4 6 27 is a matrix with 3 rows and 5 columns
More information3 Matrix Algebra. 3.1 Operations on matrices
3 Matrix Algebra A matrix is a rectangular array of numbers; it is of size m n if it has m rows and n columns. A 1 n matrix is a row vector; an m 1 matrix is a column vector. For example: 1 5 3 5 3 5 8
More information3 (Maths) Linear Algebra
3 (Maths) Linear Algebra References: Simon and Blume, chapters 6 to 11, 16 and 23; Pemberton and Rau, chapters 11 to 13 and 25; Sundaram, sections 1.3 and 1.5. The methods and concepts of linear algebra
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 informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS
More informationLinear Algebra (Review) Volker Tresp 2017
Linear Algebra (Review) Volker Tresp 2017 1 Vectors k is a scalar (a number) c is a column vector. Thus in two dimensions, c = ( c1 c 2 ) (Advanced: More precisely, a vector is defined in a vector space.
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationIntroduction to Matrices
214 Analysis and Design of Feedback Control Systems Introduction to Matrices Derek Rowell October 2002 Modern system dynamics is based upon a matrix representation of the dynamic equations governing the
More informationMath113: Linear Algebra. Beifang Chen
Math3: Linear Algebra Beifang Chen Spring 26 Contents Systems of Linear Equations 3 Systems of Linear Equations 3 Linear Systems 3 2 Geometric Interpretation 3 3 Matrices of Linear Systems 4 4 Elementary
More informationMath 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures
More informationComputational math: Assignment 1
Computational math: Assignment 1 Thanks Ting Gao for her Latex file 11 Let B be a 4 4 matrix to which we apply the following operations: 1double column 1, halve row 3, 3add row 3 to row 1, 4interchange
More information7. Symmetric Matrices and Quadratic Forms
Linear Algebra 7. Symmetric Matrices and Quadratic Forms CSIE NCU 1 7. Symmetric Matrices and Quadratic Forms 7.1 Diagonalization of symmetric matrices 2 7.2 Quadratic forms.. 9 7.4 The singular value
More informationMATRICES AND ITS APPLICATIONS
MATRICES AND ITS Elementary transformations and elementary matrices Inverse using elementary transformations Rank of a matrix Normal form of a matrix Linear dependence and independence of vectors APPLICATIONS
More informationCS 143 Linear Algebra Review
CS 143 Linear Algebra Review Stefan Roth September 29, 2003 Introductory Remarks This review does not aim at mathematical rigor very much, but instead at ease of understanding and conciseness. Please see
More informationPreliminary Linear Algebra 1. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 100
Preliminary Linear Algebra 1 Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 100 Notation for all there exists such that therefore because end of proof (QED) Copyright c 2012
More information1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)
1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
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 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 information