Linear Least Squares. Using SVD Decomposition.
|
|
- Norah Stanley
- 5 years ago
- Views:
Transcription
1 Linear Least Squares. Using SVD Decomposition. Dmitriy Leykekhman Spring 2011 Goals SVD-decomposition. Solving LLS with SVD-decomposition. D. Leykekhman Linear Least Squares 1
2 SVD Decomposition. For any matrix A R m n there exist orthogonal matrices U R m m, V R n n and a diagonal matrix Σ R m n, i.e., σ σr Σ = 0 for m n with diagonal entries such that A = UΣV T σ 1 σ r > σ r+1 = = σ min {m,n} = 0 D. Leykekhman Linear Least Squares 2
3 SVD Decomposition. For any matrix A R m n there exist orthogonal matrices U R m m, V R n n and a diagonal matrix Σ R m n, i.e., σ 1... σ r 0 Σ =... for m n with diagonal entries such that A = UΣV T σ 1 σ r > σ r+1 = = σ min {m,n} = 0 D. Leykekhman Linear Least Squares 2
4 SVD Decomposition. The decomposition A = UΣV T is called Singular Value Decomposition (SVD). It is very important decomposition of a matrix and tells us a lot about its structure. It can be computed using the Matlab command svd. The diagonal entries of Σ are called the singular values of A. The columns of U are called left singular vectors and the columns of V are called right singular vectors. D. Leykekhman Linear Least Squares 3
5 SVD Decomposition. The decomposition A = UΣV T is called Singular Value Decomposition (SVD). It is very important decomposition of a matrix and tells us a lot about its structure. It can be computed using the Matlab command svd. The diagonal entries of Σ are called the singular values of A. The columns of U are called left singular vectors and the columns of V are called right singular vectors. Using the orthogonality of V we can write it in the form AV = UΣ We can interpret it as follows: there exists a special orthonormal set of vectors (i.e. the columns of V ), that is mapped by the matrix A into an orthonormal set of vectors (i.e. the columns of U). D. Leykekhman Linear Least Squares 3
6 Applications of SVD Decomposition. Given the SVD-Decomposition of A, A = UΣV T with σ 1 σ r > σ r+1 = = σ min {m,n} = 0 one may conclude the following: rank(a) = r, R(A) = R([u 1,..., u r ]), N(A) = R([v r+1,..., v n ]), R(A T ) = R([v 1,..., v r ]), N(A T ) = R([u r+1,..., u m ]). D. Leykekhman Linear Least Squares 4
7 Applications of SVD Decomposition. Moreover if we denote U r = [u 1,..., u r ], Σ r = diag(σ 1,..., σ r ), V r = [v 1,..., v r ], then we have A = U r Σ r V T r = u i vi T This is called the dyadic decomposition of A, decomposes the matrix A of rank r into sum of r matrices of rank 1. D. Leykekhman Linear Least Squares 5
8 Applications of SVD Decomposition. The 2-norm and the Frobenius norm of A can be easily computed from the SVD decomposition Ax 2 A 2 = sup = σ 1 x 0 x 2 m n A F = a 2 ij = σ σ2 p, j=1 p = min {m, n}. From the SVD decomposition of A it also follows that A T A = V Σ T ΣV T and AA T = UΣΣ T U T. Thus, σi 2, i = 1,..., p are the eigenvalues of symmetric matrices A T A and AA T and v i and u i are the corresponding eigenvectors. D. Leykekhman Linear Least Squares 6
9 Applications of SVD Decomposition. Theorem Let the SVD of A R m n be given by with r = rank(a). If k < r then A = U r Σ r V T r = A k = u i vi T k u i vi T, min A D 2 = A A k 2 = σ k+1, rank(d)=k and min A D p F = A A k F = σi 2, rank(d)=k k+1 D. Leykekhman Linear Least Squares 7 p = min {m, n}.
10 Solving LLS with SVD Decomposition. Consider the LLS min x Ax b 2 2 Let A = UΣV T be the SVD of A R m n. Using the orthogonality of U and V we have Ax b 2 2 = U T (AV V T x b) 2 2 = Σ V }{{} T x U T b) 2 2 = ( z i u T i b) 2 + =z m (u T i b) 2. i=r+1 D. Leykekhman Linear Least Squares 8
11 Solving LLS with SVD Decomposition. Thus, min x Ax b 2 2 = ( z i u T i b) 2 + m (u T i b) 2. i=r+1 D. Leykekhman Linear Least Squares 9
12 Solving LLS with SVD Decomposition. Thus, min x Ax b 2 2 = The solution is given ( z i u T i b) 2 + m (u T i b) 2. i=r+1 z i = ut i b, i = 1,..., r, z i = arbitrary, i = r + 1,..., n. D. Leykekhman Linear Least Squares 9
13 Solving LLS with SVD Decomposition. Thus, min x Ax b 2 2 = The solution is given ( z i u T i b) 2 + m (u T i b) 2. i=r+1 z i = ut i b, i = 1,..., r, z i = arbitrary, i = r + 1,..., n. As a result min x Ax b 2 2 = m (u T i b) 2. i=r+1 D. Leykekhman Linear Least Squares 9
14 Solving LLS with SVD Decomposition. Recall that z = V T x. Since V is orthogonal, we find that x 2 = V V T x 2 = V T x 2 = z 2. All solutions of the linear least squares problem are given by z = V T x with z i = ut i b, i = 1,..., r, z i = arbitrary, i = r + 1,..., n. D. Leykekhman Linear Least Squares 10
15 Solving LLS with SVD Decomposition. Minimum norm solution The minimum norm solution of the linear least squares problem is given by x = V z, where z R n is the vector with entries z i = ut i b, i = 1,..., r, z i The minimum norm solution is = 0, i = r + 1,..., n. x = u T i b v i D. Leykekhman Linear Least Squares 11
16 Solving LLS with SVD Decomposition. MATLAB code. % compute the SVD: [U,S,V] = svd(a); s = diag(s); % determine the effective rank r of A using singular values r = 1; while( r < size(a,2) & s(r+1) >= max(size(a))*eps*s(1) ) r = r+1; end d = U *b; x = V* ( [d(1:r)./s(1:r); zeros(n-r,1) ] ); D. Leykekhman Linear Least Squares 12
17 Conditioning of a Linear Least Squares Problem. Suppose that the data b are b = b ex + δb, where δb represents the measurement error. The minimum norm solution of min Ax (b ex + δb) 2 2 is x = u T i b v i = ( u T i b + ut i δb ) v i. D. Leykekhman Linear Least Squares 13
18 Conditioning of a Linear Least Squares Problem. Suppose that the data b are b = b ex + δb, where δb represents the measurement error. The minimum norm solution of min Ax (b ex + δb) 2 2 is x = u T i b v i = ( u T i b + ut i δb ) v i. If a singular value is small, then ut i (δb) could be large, even if u T i (δb) is small. This shows that errors δb in the data can be magnified by small singular values. D. Leykekhman Linear Least Squares 13
19 Conditioning of a Linear Least Squares Problem. % Compute A t = 10.^(0:-1:-10) ; A = [ ones(size(t)) t t.^2 t.^3 t.^4 t.^5]; % compute SVD of A [U,S,V] = svd(a); sigma = diag(s); % compute exact data xex = ones(6,1); bex = A*xex; for i = 1:10 % data perturbation deltab = 10^(-i)*(0.5-rand(size(bex))).*bex; b = bex+deltab; % solution of perturbed linear least squares problem w = U *b; x = V * (w(1:6)./ sigma); errx(i+1) = norm(x - xex); errb(i+1) = norm(deltab); end loglog(errb,errx, * ); ylabel( x^{ex} - x _2 ); xlabel( \delta b _2 ) D. Leykekhman Linear Least Squares 14
20 Conditioning of a Linear Least Squares Problem. The singular values of A in the above Matlab example are: σ σ σ σ σ σ The error x ex x 2 for different values of δb 2 (loglog-scale): We see that small perturbations δb in the measurements can lead to large errors in the solution x of the linear least squares problem if the singular values of A are small. D. Leykekhman Linear Least Squares 15
MATH 3795 Lecture 10. Regularized Linear Least Squares.
MATH 3795 Lecture 10. Regularized Linear Least Squares. Dmitriy Leykekhman Fall 2008 Goals Understanding the regularization. D. Leykekhman - MATH 3795 Introduction to Computational Mathematics Linear Least
More informationMARN 5898 Regularized Linear Least Squares.
MARN 5898 Regularized Linear Least Squares. Dmitriy Leykekhman Spring 2010 Goals Understanding the regularization. D. Leykekhman - MARN 5898 Parameter estimation in marine sciences Linear Least Squares
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 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 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 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 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 non-zero solution If Ax = λx
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 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 informationLinear Algebra, part 3. Going back to least squares. Mathematical Models, Analysis and Simulation = 0. a T 1 e. a T n e. Anna-Karin Tornberg
Linear Algebra, part 3 Anna-Karin 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 informationMATH 612 Computational methods for equation solving and function minimization Week # 2
MATH 612 Computational methods for equation solving and function minimization Week # 2 Instructor: Francisco-Javier Pancho Sayas Spring 2014 University of Delaware FJS MATH 612 1 / 38 Plan for this week
More informationThe 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 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 October 17, 005 Lecture 3 3 he Singular Value Decomposition
More information14 Singular Value Decomposition
14 Singular Value Decomposition For any high-dimensional data analysis, one s first thought should often be: can I use an SVD? The singular value decomposition is an invaluable analysis tool for dealing
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 informationThe Singular Value Decomposition
The Singular Value Decomposition An Important topic in NLA Radu Tiberiu Trîmbiţaş Babeş-Bolyai University February 23, 2009 Radu Tiberiu Trîmbiţaş ( Babeş-Bolyai University)The Singular Value Decomposition
More informationLinear Algebra, part 3 QR and SVD
Linear Algebra, part 3 QR and SVD Anna-Karin 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 informationEE263: Introduction to Linear Dynamical Systems Review Session 9
EE63: Introduction to Linear Dynamical Systems Review Session 9 SVD continued EE63 RS9 1 Singular Value Decomposition recall any nonzero matrix A R m n, with Rank(A) = r, has an SVD given by A = UΣV T,
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 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 2011-2012 A square matrix is symmetric if A T
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 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 informationNumerical Methods I Singular Value Decomposition
Numerical Methods I Singular Value Decomposition Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 October 9th, 2014 A. Donev (Courant Institute)
More information15 Singular Value Decomposition
15 Singular Value Decomposition For any high-dimensional data analysis, one s first thought should often be: can I use an SVD? The singular value decomposition is an invaluable analysis tool for dealing
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 information1 Linearity and Linear Systems
Mathematical Tools for Neuroscience (NEU 34) Princeton University, Spring 26 Jonathan Pillow Lecture 7-8 notes: Linear systems & SVD Linearity and Linear Systems Linear system is a kind of mapping f( x)
More informationLeast Squares. Tom Lyche. October 26, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Least Squares Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo October 26, 2010 Linear system Linear system Ax = b, A C m,n, b C m, x C n. under-determined
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 informationMatrix decompositions
Matrix decompositions Zdeněk Dvořák May 19, 2015 Lemma 1 (Schur decomposition). If A is a symmetric real matrix, then there exists an orthogonal matrix Q and a diagonal matrix D such that A = QDQ T. The
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More informationBasic Calculus Review
Basic Calculus Review Lorenzo Rosasco ISML Mod. 2 - Machine Learning Vector Spaces Functionals and Operators (Matrices) Vector Space A vector space is a set V with binary operations +: V V V and : R V
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: Face Recognition and Feature Reduction
Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab Lecture 11-1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed
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 informationAlgorithmen zur digitalen Bildverarbeitung I
ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG INSTITUT FÜR INFORMATIK Lehrstuhl für Mustererkennung und Bildverarbeitung Prof. Dr.-Ing. Hans Burkhardt Georges-Köhler-Allee Geb. 05, Zi 01-09 D-79110 Freiburg Tel.
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 informationSINGULAR VALUE DECOMPOSITION
DEPARMEN OF MAHEMAICS ECHNICAL REPOR SINGULAR VALUE DECOMPOSIION Andrew Lounsbury September 8 No 8- ENNESSEE ECHNOLOGICAL UNIVERSIY Cookeville, N 38 Singular Value Decomposition Andrew Lounsbury Department
More informationPseudoinverse & Orthogonal Projection Operators
Pseudoinverse & Orthogonal Projection Operators ECE 174 Linear & Nonlinear Optimization Ken Kreutz-Delgado ECE Department, UC San Diego Ken Kreutz-Delgado (UC San Diego) ECE 174 Fall 2016 1 / 48 The Four
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 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 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 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 informationLecture: Face Recognition and Feature Reduction
Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab 1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed in the
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 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 informationThroughout these notes we assume V, W are finite dimensional inner product spaces over C.
Math 342 - Linear Algebra II Notes Throughout these notes we assume V, W are finite dimensional inner product spaces over C 1 Upper Triangular Representation Proposition: Let T L(V ) There exists an orthonormal
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 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 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.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
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 2-6 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 positive-definite
More informationProbabilistic Latent Semantic Analysis
Probabilistic Latent Semantic Analysis Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr
More information2. Review of Linear Algebra
2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear
More informationσ 11 σ 22 σ pp 0 with p = min(n, m) The σ ii s are the singular values. Notation change σ ii A 1 σ 2
HE SINGULAR VALUE DECOMPOSIION he SVD existence - properties. Pseudo-inverses and the SVD Use of SVD for least-squares problems Applications of the SVD he Singular Value Decomposition (SVD) heorem For
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Numerical Linear Algebra Background Cho-Jui Hsieh UC Davis May 15, 2018 Linear Algebra Background Vectors A vector has a direction and a magnitude
More informationSingular Value Decomposition (SVD)
School of Computing National University of Singapore CS CS524 Theoretical Foundations of Multimedia More Linear Algebra Singular Value Decomposition (SVD) The highpoint of linear algebra Gilbert Strang
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 information8. the singular value decomposition
8. the singular value decomposition cmda 3606; mark embree version of 19 February 2017 The singular value decomposition (SVD) is among the most important and widely applicable matrix factorizations. It
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science : Dynamic Systems Spring 2011
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 64: Dynamic Systems Spring Homework 4 Solutions Exercise 47 Given a complex square matrix A, the definition
More informationLecture 2: Linear Algebra Review
EE 227A: Convex Optimization and Applications January 19 Lecture 2: Linear Algebra Review Lecturer: Mert Pilanci Reading assignment: Appendix C of BV. Sections 2-6 of the web textbook 1 2.1 Vectors 2.1.1
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 informationNumerical Methods. Elena loli Piccolomini. Civil Engeneering. piccolom. Metodi Numerici M p. 1/??
Metodi Numerici M p. 1/?? Numerical Methods Elena loli Piccolomini Civil Engeneering http://www.dm.unibo.it/ piccolom elena.loli@unibo.it Metodi Numerici M p. 2/?? Least Squares Data Fitting Measurement
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 informationPositive Definite Matrix
1/29 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Positive Definite, Negative Definite, Indefinite 2/29 Pure Quadratic Function
More informationEigenvectors and SVD 1
Eigenvectors and SVD 1 Definition Eigenvectors of a square matrix Ax=λx, x=0. Intuition: x is unchanged by A (except for scaling) Examples: axis of rotation, stationary distribution of a Markov chain 2
More informationMATH 350: Introduction to Computational Mathematics
MATH 350: Introduction to Computational Mathematics Chapter V: Least Squares Problems Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Spring 2011 fasshauer@iit.edu MATH
More informationBindel, Fall 2009 Matrix Computations (CS 6210) Week 8: Friday, Oct 17
Logistics Week 8: Friday, Oct 17 1. HW 3 errata: in Problem 1, I meant to say p i < i, not that p i is strictly ascending my apologies. You would want p i > i if you were simply forming the matrices and
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 informationHomework 1. Yuan Yao. September 18, 2011
Homework 1 Yuan Yao September 18, 2011 1. Singular Value Decomposition: The goal of this exercise is to refresh your memory about the singular value decomposition and matrix norms. A good reference to
More informationSingular Value Decomposition
Chapter 6 Singular Value Decomposition In Chapter 5, we derived a number of algorithms for computing the eigenvalues and eigenvectors of matrices A R n n. Having developed this machinery, we complete our
More informationLinear Algebra Methods for Data Mining
Linear Algebra Methods for Data Mining Saara Hyvönen, Saara.Hyvonen@cs.helsinki.fi Spring 2007 1. Basic Linear Algebra Linear Algebra Methods for Data Mining, Spring 2007, University of Helsinki Example
More informationThe University of Texas at Austin Department of Electrical and Computer Engineering. EE381V: Large Scale Learning Spring 2013.
The University of Texas at Austin Department of Electrical and Computer Engineering EE381V: Large Scale Learning Spring 2013 Assignment Two Caramanis/Sanghavi Due: Tuesday, Feb. 19, 2013. Computational
More informationPrincipal Component Analysis
Machine Learning Michaelmas 2017 James Worrell Principal Component Analysis 1 Introduction 1.1 Goals of PCA Principal components analysis (PCA) is a dimensionality reduction technique that can be used
More informationIntroduction to SVD and Applications
Introduction to SVD and Applications Eric Kostelich and Dave Kuhl MSRI Climate Change Summer School July 18, 2008 Introduction The goal of this exercise is to familiarize you with the basics of the singular
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Lecture 5: Numerical Linear Algebra Cho-Jui Hsieh UC Davis April 20, 2017 Linear Algebra Background Vectors A vector has a direction and a magnitude
More informationLinGloss. A glossary of linear algebra
LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasi-triangular A matrix A is quasi-triangular iff it is a triangular matrix except its diagonal
More informationLatent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology
Latent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology M. Soleymani Fall 2014 Most slides have been adapted from: Profs. Manning, Nayak & Raghavan (CS-276,
More informationLecture 5 Singular value decomposition
Lecture 5 Singular value decomposition Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University, tieli@pku.edu.cn
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 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 informationThe QR Decomposition
The QR Decomposition We have seen one major decomposition of a matrix which is A = LU (and its variants) or more generally PA = LU for a permutation matrix P. This was valid for a square matrix and aided
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 informationNORMS ON SPACE OF MATRICES
NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system
More informationLatent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology
Latent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology M. Soleymani Fall 2016 Most slides have been adapted from: Profs. Manning, Nayak & Raghavan (CS-276,
More informationVector and Matrix Norms. Vector and Matrix Norms
Vector and Matrix Norms Vector Space Algebra Matrix Algebra: We let x x and A A, where, if x is an element of an abstract vector space n, and A = A: n m, then x is a complex column vector of length n whose
More informationLinear Algebra in Actuarial Science: Slides to the lecture
Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations
More informationIV. Matrix Approximation using Least-Squares
IV. Matrix Approximation using Least-Squares The SVD and Matrix Approximation We begin with the following fundamental question. Let A be an M N matrix with rank R. What is the closest matrix to A that
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 informationThe Singular Value Decomposition
CHAPTER 6 The Singular Value Decomposition Exercise 67: SVD examples (a) For A =[, 4] T we find a matrixa T A =5,whichhastheeigenvalue =5 Thisprovidesuswiththesingularvalue =+ p =5forA Hence the matrix
More informationSection 3.9. Matrix Norm
3.9. Matrix Norm 1 Section 3.9. Matrix Norm Note. We define several matrix norms, some similar to vector norms and some reflecting how multiplication by a matrix affects the norm of a vector. We use matrix
More informationSingular Value Decomposition
Singular Value Decomposition (Com S 477/577 Notes Yan-Bin Jia Sep, 7 Introduction Now comes a highlight of linear algebra. Any real m n matrix can be factored as A = UΣV T where U is an m m orthogonal
More informationMath 515 Fall, 2008 Homework 2, due Friday, September 26.
Math 515 Fall, 2008 Homework 2, due Friday, September 26 In this assignment you will write efficient MATLAB codes to solve least squares problems involving block structured matrices known as Kronecker
More informationDesigning Information Devices and Systems II
EECS 16B Fall 2016 Designing Information Devices and Systems II Linear Algebra Notes Introduction In this set of notes, we will derive the linear least squares equation, study the properties symmetric
More information7 Principal Component Analysis
7 Principal Component Analysis This topic will build a series of techniques to deal with high-dimensional data. Unlike regression problems, our goal is not to predict a value (the y-coordinate), it is
More informationON THE SINGULAR DECOMPOSITION OF MATRICES
An. Şt. Univ. Ovidius Constanţa Vol. 8, 00, 55 6 ON THE SINGULAR DECOMPOSITION OF MATRICES Alina PETRESCU-NIŢǍ Abstract This paper is an original presentation of the algorithm of the singular decomposition
More informationLinear Algebra Methods for Data Mining
Linear Algebra Methods for Data Mining Saara Hyvönen, Saara.Hyvonen@cs.helsinki.fi Spring 2007 The Singular Value Decomposition (SVD) continued Linear Algebra Methods for Data Mining, Spring 2007, University
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 informationMIT Final Exam Solutions, Spring 2017
MIT 8.6 Final Exam Solutions, Spring 7 Problem : For some real matrix A, the following vectors form a basis for its column space and null space: C(A) = span,, N(A) = span,,. (a) What is the size m n of
More informationTHE SINGULAR VALUE DECOMPOSITION MARKUS GRASMAIR
THE SINGULAR VALUE DECOMPOSITION MARKUS GRASMAIR 1. Definition Existence Theorem 1. Assume that A R m n. Then there exist orthogonal matrices U R m m V R n n, values σ 1 σ 2... σ p 0 with p = min{m, n},
More informationReview of similarity transformation and Singular Value Decomposition
Review of similarity transformation and Singular Value Decomposition Nasser M Abbasi Applied Mathematics Department, California State University, Fullerton July 8 7 page compiled on June 9, 5 at 9:5pm
More informationLinear Algebra Review. Fei-Fei Li
Linear Algebra Review Fei-Fei 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 information