The Singular Value Decomposition


 Verity Randall
 1 years ago
 Views:
Transcription
1 The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall / 13
2 Review of Key Concepts We review some key definitions and results about matrices that will be used in this section. The transpose of a matrix A, denoted A T is the matrix obtained from A by switching its rows and columns. In other words, if A = (a ij ) then A T = (a ji ). The conjugate transpose of a matrix A, denoted A is obtained from A by switching its rows and columns and taking the conjugate of its entries. In other words, if A = (a ij ) then A = (a ji ). A matrix A is said to be symmetric if A = A T. Symmetric matrices have the following properties: Their eigenvalues are always real. They are always diagonalizable. Their eigenvectors are orthogonal. They are orthogonally diagonalizable that is if A is such a matrix then there exists an orthogonal matrix P such that P 1 AP is diagonal. Philippe B. Laval (KSU) SVD Fall / 13
3 Review of Key Concepts A matrix A is said to be Hermitian if A = A. For matrices with real entries, being Hermitian is the same as being symmetric. An n n matrix A is said to be normal if A A = AA. Obviously, Hermitian matrices are also normal. A matrix A is said to be unitary if AA = A A = I. Unitary matrices have the following properties: They preserve the dot product that is Ax, Ay = x, y Their columns and rows are orthogonal. They are always diagonalizable. det A = 1. A 1 = A A matrix A is said to be orthogonal if AA T = A T A = I. Orthogonal matrices have the following properties: They preserve the dot product that is Ax, Ay = x, y Their columns and rows are orthogonal. det A = 1. A 1 = A T Philippe B. Laval (KSU) SVD Fall / 13
4 Review of Key Concepts A quadratic form on R n is a function Q defined on R n by Q (x) = x T Ax for some n n matrix A. Here are a few important facts about quadratic forms: In the case A is symmetric, there exists a change of variable x = Py that transforms x T Ax into y T Dy where D is a diagonal matrix. In the case A is symmetric, the maximum value of x T Ax is the absolute value of the largest eigenvalue λ 1 of A and it happens in the direction of u 1 the corresponding eigenvector. Philippe B. Laval (KSU) SVD Fall / 13
5 Introduction to the SVD of a Matrix Recall that if A is symmetric, then its eigenvalues are real. Moreover, if Ax = λx and x = 1 then Ax = λx = λ x = λ. Hence, λ measures the amount by which A stretches (or shrinks) vectors which have the same direction as the eigenvectors. If λ 1 is the eigenvalue with largest magnitude and v 1 is its corresponding eigenvectors, then v 1 gives the direction in which the stretching effect of A is the greatest. This description of v 1 and λ 1 has an analogue for rectangular matrices that will lead the the SVD. We begin with an example. Example [ ] Let A =. Find a unit vector x at which the length of Ax is maximized and compute this maximum length. Philippe B. Laval (KSU) SVD Fall / 13
6 The Singular Values of a Matrix Let A be an m n matrix. Then as noted in the example, A T A is an n n symmetric matrix hence orthogonally diagonalizable. Let {v 1, v 2,..., v n } be an orthonormal basis for R n consisting of the eigenvectors of A T A and let λ 1, λ 2,..., λ n be the corresponding eigenvalues of A T A. Then, for 1 i n, we have: Av i 2 = λ i we see that all the eigenvalues of A T A are nonnegative. by renumbering, we may assume that λ 1 λ 2... λ n. Definition The singular values of A are the square roots of the eigenvalues λ i of A T A, denoted σ i and they are arranged in decreasing order. In other words, σ i = λ i Since Av i 2 = λ i, we see that σ i is the length of the vectors Av i, where v i are the eigenvectors of A T A. Philippe B. Laval (KSU) SVD Fall / 13
7 The Singular Values of a Matrix Example Find the singular values of A, the matrix of the previous example. We have the following important theorem. Theorem Suppose that {v 1, v 2,..., v n } is an orthonormal basis for R n consisting of the eigenvectors of A T A arranged so that the corresponding eigenvalues of A T A satisfy λ 1 λ 2... λ n, and suppose that A has r nonzero singular values. Then, {Av 1, Av 2,..., Av r } is an orthogonal basis for ColA, and ranka = r. Philippe B. Laval (KSU) SVD Fall / 13
8 The SVD of an mxn Matrix Let A be an m n matrix with r nonzero singular values where r min (m, n). Define D to be the r r diagonal matrix consisting of these r nonzero singular values of A such that σ 1 σ 2... σ r. Let [ ] D 0 Σ = 0 0 be an m n matrix. The SVD decomposition of A will involve Σ. More specifically, we have the following theorem. Theorem Let A be an m n matrix with rank r. Then there exists an m n matrix Σ as in 1 as well as an m m orthogonal matrix U and an n n orthogonal matrix V such that A = UΣV T (1) Philippe B. Laval (KSU) SVD Fall / 13
9 The SVD of an mxn Matrix Definition Any decomposition A = UΣV T with U and V orthogonal, Σ as in 1 and positive diagonal entries for D,is called a singular value decomposition (SVD) of A. The matrix U and V are not uniquely determined, but the diagonal entries of Σ are necessarily the singular values of A. The columns of U are called the left singular vectors of A and the columns of V are called the right singular vectors of A. Example Find the SVD of the matrix A in the examples above. Philippe B. Laval (KSU) SVD Fall / 13
10 The SVD of an mxn Matrix We outline the proof. Let λ i and v i be as above. Then {Av 1, Av 2,..., Av r } is an orthogonal basis for col A. We normalize each Av i to obtain an orthonormal basis {u 1, u 2,..., u r }, where u i = 1 Av i Av i = 1 σ i Av i thus Av i = σ i u i (1 i r). Next, we extend {u 1, u 2,..., u r } to an orthonormal basis {u 1, u 2,..., u m } of R m and let U = [u 1, u 2,..., u m ] and V = [v 1, v 2,..., v n ]. By constructions, both U and V are orthogonal matrices. Also, AV = [σ 1 u 1 σ 2 u 2 σ r u r 0 0] Philippe B. Laval (KSU) SVD Fall / 13
11 The SVD of an mxn Matrix Let D and Σ be as above, then σ σ UΣ = [u 1, u 2,..., u m ] 0 0 σ r = [σ 1 u 1 σ 2 u 2 σ r u r 0 0] = AV Therefore, UΣV T = AVV T = A since V is orthogonal (VV T = I ) Philippe B. Laval (KSU) SVD Fall / 13
12 The SVD and MATLAB The MATLAB command is svd. Two useful formats for this command are: 1 X=svd(A) will returns a vector X containing the singular values of A. 2 [U,S,V] = svd(x) produces a diagonal matrix S, of the same dimension as X and with nonnegative diagonal elements in decreasing order, and unitary matrices U and V so that X = USV T. Philippe B. Laval (KSU) SVD Fall / 13
13 Exercises See the problems at the end of the notes on the basics of SVD. Philippe B. Laval (KSU) SVD Fall / 13
The Singular Value Decomposition (SVD) and Principal Component Analysis (PCA)
Chapter 5 The Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) 5.1 Basics of SVD 5.1.1 Review of Key Concepts We review some key definitions and results about matrices that will
More informationMaths for Signals and Systems Linear Algebra in Engineering
Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE
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 informationChapter 7: Symmetric Matrices and Quadratic Forms
Chapter 7: Symmetric Matrices and Quadratic Forms (Last Updated: December, 06) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved
More informationNotes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T.
Notes on singular value decomposition for Math 54 Recall that if A is a symmetric n n matrix, then A has real eigenvalues λ 1,, λ n (possibly repeated), and R n has an orthonormal basis v 1,, v n, where
More informationUNIT 6: The singular value decomposition.
UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 20112012 A square matrix is symmetric if A T
More informationSingular Value Decomposition
Singular Value Decomposition Motivatation The diagonalization theorem play a part in many interesting applications. Unfortunately not all matrices can be factored as A = PDP However a factorization A =
More informationIndependent Component Analysis
Independent Component Analysis Philippe B. Laval KSU Fall 2017 Philippe B. Laval (KSU) ICA Fall 2017 1 / 18 Introduction Independent Component Analysis (ICA) falls under the broader topic of Blind Source
More informationProperties of Matrices and Operations on Matrices
Properties of Matrices and Operations on Matrices A common data structure for statistical analysis is a rectangular array or matris. Rows represent individual observational units, or just observations,
More informationNumerical Linear Algebra
Numerical Linear Algebra Direct Methods Philippe B. Laval KSU Fall 2017 Philippe B. Laval (KSU) Linear Systems: Direct Solution Methods Fall 2017 1 / 14 Introduction The solution of linear systems is one
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 informationNumerical Linear Algebra Homework Assignment  Week 2
Numerical Linear Algebra Homework Assignment  Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.
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 informationMath 315: Linear Algebra Solutions to Assignment 7
Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are
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 informationLinear Least Squares. Using SVD Decomposition.
Linear Least Squares. Using SVD Decomposition. Dmitriy Leykekhman Spring 2011 Goals SVDdecomposition. Solving LLS with SVDdecomposition. D. Leykekhman Linear Least Squares 1 SVD Decomposition. For any
More informationNotes on Eigenvalues, Singular Values and QR
Notes on Eigenvalues, Singular Values and QR Michael Overton, Numerical Computing, Spring 2017 March 30, 2017 1 Eigenvalues Everyone who has studied linear algebra knows the definition: given a square
More informationLinear Algebra Review. FeiFei Li
Linear Algebra Review FeiFei Li 1 / 51 Vectors Vectors and matrices are just collections of ordered numbers that represent something: movements in space, scaling factors, pixel brightnesses, etc. A vector
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. FeiFei Li
Linear Algebra Review FeiFei Li 1 / 37 Vectors Vectors and matrices are just collections of ordered numbers that represent something: movements in space, scaling factors, pixel brightnesses, etc. A vector
More informationHomework 1 Elena Davidson (B) (C) (D) (E) (F) (G) (H) (I)
CS 106 Spring 2004 Homework 1 Elena Davidson 8 April 2004 Problem 1.1 Let B be a 4 4 matrix to which we apply the following operations: 1. double column 1, 2. halve row 3, 3. add row 3 to row 1, 4. interchange
More informationComputational Methods CMSC/AMSC/MAPL 460. Eigenvalues and Eigenvectors. Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 Eigenvalues and Eigenvectors Ramani Duraiswami, Dept. of Computer Science Eigen Values of a Matrix Recap: A N N matrix A has an eigenvector x (nonzero) with corresponding
More informationLEC 2: Principal Component Analysis (PCA) A First Dimensionality Reduction Approach
LEC 2: Principal Component Analysis (PCA) A First Dimensionality Reduction Approach Dr. Guangliang Chen February 9, 2016 Outline Introduction Review of linear algebra Matrix SVD PCA Motivation The digits
More informationEcon 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms. 1 Diagonalization and Change of Basis
Econ 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms De La Fuente notes that, if an n n matrix has n distinct eigenvalues, it can be diagonalized. In this supplement, we will provide
More informationComputational Methods. Eigenvalues and Singular Values
Computational Methods Eigenvalues and Singular Values Manfred Huber 2010 1 Eigenvalues and Singular Values Eigenvalues and singular values describe important aspects of transformations and of data relations
More informationFall TMA4145 Linear Methods. Exercise set Given the matrix 1 2
Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an
More informationSummary of Week 9 B = then A A =
Summary of Week 9 Finding the square root of a positive operator Last time we saw that positive operators have a unique positive square root We now briefly look at how one would go about calculating the
More information18.06 Problem Set 8  Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8  Solutions Due Wednesday, 4 November 2007 at 4 pm in 206 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
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 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 informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall 2015 1 / 14 Introduction We define eigenvalues and eigenvectors. We discuss how to
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 informationNumerical Methods I Singular Value Decomposition
Numerical Methods I Singular Value Decomposition Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATHGA 2011.003 / CSCIGA 2945.003, Fall 2014 October 9th, 2014 A. Donev (Courant Institute)
More informationSingular Value Decomposition
Chapter 5 Singular Value Decomposition We now reach an important Chapter in this course concerned with the Singular Value Decomposition of a matrix A. SVD, as it is commonly referred to, is one of the
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 informationComputational Methods CMSC/AMSC/MAPL 460. EigenValue decomposition Singular Value Decomposition. Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 EigenValue decomposition Singular Value Decomposition Ramani Duraiswami, Dept. of Computer Science Hermitian Matrices A square matrix for which A = A H is said
More information(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =
. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(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 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
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 information1 Linearity and Linear Systems
Mathematical Tools for Neuroscience (NEU 34) Princeton University, Spring 26 Jonathan Pillow Lecture 78 notes: Linear systems & SVD Linearity and Linear Systems Linear system is a kind of mapping f( x)
More informationQuadratic forms. Here. Thus symmetric matrices are diagonalizable, and the diagonalization can be performed by means of an orthogonal matrix.
Quadratic forms 1. Symmetric matrices An n n matrix (a ij ) n ij=1 with entries on R is called symmetric if A T, that is, if a ij = a ji for all 1 i, j n. We denote by S n (R) the set of all n n symmetric
More informationLinGloss. A glossary of linear algebra
LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasitriangular A matrix A is quasitriangular iff it is a triangular matrix except its diagonal
More informationThe Singular Value Decomposition
The Singular Value Decomposition An Important topic in NLA Radu Tiberiu Trîmbiţaş BabeşBolyai University February 23, 2009 Radu Tiberiu Trîmbiţaş ( BabeşBolyai University)The Singular Value Decomposition
More informationThe Singular Value Decomposition and Least Squares Problems
The Singular Value Decomposition and Least Squares Problems Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 27, 2009 Applications of SVD solving
More informationDimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas
Dimensionality Reduction: PCA Nicholas Ruozzi University of Texas at Dallas Eigenvalues λ is an eigenvalue of a matrix A R n n if the linear system Ax = λx has at least one nonzero solution If Ax = λx
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 informationforms Christopher Engström November 14, 2014 MAA704: Matrix factorization and canonical forms Matrix properties Matrix factorization Canonical forms
Christopher Engström November 14, 2014 Hermitian LU QR echelon Contents of todays lecture Some interesting / useful / important of matrices Hermitian LU QR echelon Rewriting a as a product of several matrices.
More informationbe a Householder matrix. Then prove the followings H = I 2 uut Hu = (I 2 uu u T u )u = u 2 uut u
MATH 434/534 Theoretical Assignment 7 Solution Chapter 7 (71) Let H = I 2uuT Hu = u (ii) Hv = v if = 0 be a Householder matrix Then prove the followings H = I 2 uut Hu = (I 2 uu )u = u 2 uut u = u 2u =
More informationAssignment #10: Diagonalization of Symmetric Matrices, Quadratic Forms, Optimization, Singular Value Decomposition. Name:
Assignment #10: Diagonalization of Symmetric Matrices, Quadratic Forms, Optimization, Singular Value Decomposition Due date: Friday, May 4, 2018 (1:35pm) Name: Section Number Assignment #10: Diagonalization
More informationApplied Mathematics 205. Unit V: Eigenvalue Problems. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit V: Eigenvalue Problems Lecturer: Dr. David Knezevic Unit V: Eigenvalue Problems Chapter V.2: Fundamentals 2 / 31 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors of
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 informationProblem # Max points possible Actual score Total 120
FINAL EXAMINATION  MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to
More informationSymmetric and anti symmetric matrices
Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal
More informationLinear Algebra (Review) Volker Tresp 2018
Linear Algebra (Review) Volker Tresp 2018 1 Vectors k, M, N are scalars A onedimensional array c is a column vector. Thus in two dimensions, ( ) c1 c = c 2 c i is the ith component of c c T = (c 1, c
More informationSingular value decomposition
Singular value decomposition The eigenvalue decomposition (EVD) for a square matrix A gives AU = UD. Let A be rectangular (m n, m > n). A singular value σ and corresponding pair of singular vectors u (m
More informationMATH36001 Generalized Inverses and the SVD 2015
MATH36001 Generalized Inverses and the SVD 201 1 Generalized Inverses of Matrices A matrix has an inverse only if it is square and nonsingular. However there are theoretical and practical applications
More informationLinear Algebra, part 3. Going back to least squares. Mathematical Models, Analysis and Simulation = 0. a T 1 e. a T n e. AnnaKarin Tornberg
Linear Algebra, part 3 AnnaKarin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2010 Going back to least squares (Sections 1.7 and 2.3 from Strang). We know from before: The vector
More informationJordan Normal Form and Singular Decomposition
University of Debrecen Diagonalization and eigenvalues Diagonalization We have seen that if A is an n n square matrix, then A is diagonalizable if and only if for all λ eigenvalues of A we have dim(u λ
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 information8. Diagonalization.
8. Diagonalization 8.1. Matrix Representations of Linear Transformations Matrix of A Linear Operator with Respect to A Basis We know that every linear transformation T: R n R m has an associated standard
More informationThe Principal Component Analysis
The Principal Component Analysis Philippe B. Laval KSU Fall 2017 Philippe B. Laval (KSU) PCA Fall 2017 1 / 27 Introduction Every 80 minutes, the two Landsat satellites go around the world, recording images
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 information1 Linear Algebra Problems
Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skewhermitian if A = A and
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 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 informationNotes on Linear Algebra
1 Notes on Linear Algebra Jean Walrand August 2005 I INTRODUCTION Linear Algebra is the theory of linear transformations Applications abound in estimation control and Markov chains You should be familiar
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2017 LECTURE 5
STAT 39: MATHEMATICAL COMPUTATIONS I FALL 17 LECTURE 5 1 existence of svd Theorem 1 (Existence of SVD) Every matrix has a singular value decomposition (condensed version) Proof Let A C m n and for simplicity
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 informationLecture notes on Quantum Computing. Chapter 1 Mathematical Background
Lecture notes on Quantum Computing Chapter 1 Mathematical Background Vector states of a quantum system with n physical states are represented by unique vectors in C n, the set of n 1 column vectors 1 For
More information1. The Polar Decomposition
A PERSONAL INTERVIEW WITH THE SINGULAR VALUE DECOMPOSITION MATAN GAVISH Part. Theory. The Polar Decomposition In what follows, F denotes either R or C. The vector space F n is an inner product space with
More informationPseudoinverse & Orthogonal Projection Operators
Pseudoinverse & Orthogonal Projection Operators ECE 174 Linear & Nonlinear Optimization Ken KreutzDelgado ECE Department, UC San Diego Ken KreutzDelgado (UC San Diego) ECE 174 Fall 2016 1 / 48 The Four
More information3D Computer Vision  WT 2004
3D Computer Vision  WT 2004 Singular Value Decomposition Darko Zikic CAMP  Chair for Computer Aided Medical Procedures November 4, 2004 1 2 3 4 5 Properties For any given matrix A R m n there exists
More informationlinearly indepedent eigenvectors as the multiplicity of the root, but in general there may be no more than one. For further discussion, assume matrice
3. Eigenvalues and Eigenvectors, Spectral Representation 3.. Eigenvalues and Eigenvectors A vector ' is eigenvector of a matrix K, if K' is parallel to ' and ' 6, i.e., K' k' k is the eigenvalue. If is
More information18.06 Problem Set 10  Solutions Due Thursday, 29 November 2007 at 4 pm in
86 Problem Set  Solutions Due Thursday, 29 November 27 at 4 pm in 26 Problem : (5=5+5+5) Take any matrix A of the form A = B H CB, where B has full column rank and C is Hermitian and positivedefinite
More informationPositive Definite Matrix
1/29 ChiaPing Chen Professor Department of Computer Science and Engineering National Sun Yatsen University Linear Algebra Positive Definite, Negative Definite, Indefinite 2/29 Pure Quadratic Function
More informationLinear Algebra, part 3 QR and SVD
Linear Algebra, part 3 QR and SVD AnnaKarin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2012 Going back to least squares (Section 1.4 from Strang, now also see section 5.2). We
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 informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tuberlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
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 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 informationBackground Mathematics (2/2) 1. David Barber
Background Mathematics (2/2) 1 David Barber University College London Modified by Samson Cheung (sccheung@ieee.org) 1 These slides accompany the book Bayesian Reasoning and Machine Learning. The book and
More informationLinear algebra and applications to graphs Part 1
Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces
More informationa 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12
24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2
More informationDIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix
DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that
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 informationCheck that your exam contains 30 multiplechoice questions, numbered sequentially.
MATH EXAM SPRING VERSION A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER On your scantron, write and bubble your PSU ID, Section Number, and Test Version. Failure to correctly code these items may result
More informationLecture 8: Linear Algebra Background
CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 8: Linear Algebra Background Lecturer: Shayan Oveis Gharan 2/1/2017 Scribe: Swati Padmanabhan Disclaimer: These notes have not been subjected
More informationLecture 15, 16: Diagonalization
Lecture 15, 16: Diagonalization Motivation: Eigenvalues and Eigenvectors are easy to compute for diagonal matrices. Hence, we would like (if possible) to convert matrix A into a diagonal matrix. Suppose
More informationFunctional Analysis Review
Outline 9.520: Statistical Learning Theory and Applications February 8, 2010 Outline 1 2 3 4 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all
More informationSingular Value Decomposition
Singular Value Decomposition CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Singular Value Decomposition 1 / 35 Understanding
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 informationMATH 583A REVIEW SESSION #1
MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Eigenvalue Problems; Similarity Transformations Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 18 Eigenvalue
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 informationLinear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4
Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am  :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary
More informationReview of some mathematical tools
MATHEMATICAL FOUNDATIONS OF SIGNAL PROCESSING Fall 2016 Benjamín Béjar Haro, Mihailo Kolundžija, Reza Parhizkar, Adam Scholefield Teaching assistants: Golnoosh Elhami, Hanjie Pan Review of some mathematical
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 informationMa/CS 6b Class 20: Spectral Graph Theory
Ma/CS 6b Class 20: Spectral Graph Theory By Adam Sheffer Eigenvalues and Eigenvectors A an n n matrix of real numbers. The eigenvalues of A are the numbers λ such that Ax = λx for some nonzero vector x
More informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More informationMath Final December 2006 C. Robinson
Math 2851 Final December 2006 C. Robinson 2 5 8 5 1 2 01 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the
More information