Preface to Second Edition... vii. Preface to First Edition...

Size: px
Start display at page:

Download "Preface to Second Edition... vii. Preface to First Edition..."

Transcription

1 Contents Preface to Second Edition vii Preface to First Edition ix Part I Linear Algebra 1 Basic Vector/Matrix Structure and Notation Vectors Arrays Matrices Representation of Data What You Compute and What You Don t Vectors and Vector Spaces Operations on Vectors Linear Combinations and Linear Independence Vector Spaces and Spaces of Vectors Basis Sets for Vector Spaces Inner Products Norms Normalized Vectors Metrics and Distances Orthogonal Vectors and Orthogonal Vector Spaces The One Vector Cartesian Coordinates and Geometrical Properties of Vectors Cartesian Geometry Projections Angles between Vectors Orthogonalization Transformations; Gram-Schmidt Orthonormal Basis Sets

2 xviii Contents Approximation of Vectors Flats, Affine Spaces, and Hyperplanes Cones Cross Products in IR Centered Vectors and Variances and Covariances of Vectors The Mean and Centered Vectors The Standard Deviation, the Variance, andscaled Vectors Covariances and Correlations between Vectors Exercises Basic Properties of Matrices Basic Definitions and Notation Matrix Shaping Operators Partitioned Matrices Matrix Addition Scalar-Valued Operators on Square Matrices:The Trace Scalar-Valued Operators on Square Matrices:The Determinant Multiplication of Matrices and Multiplication ofvectors and Matrices Matrix Multiplication (Cayley) Multiplication of Matrices with Special Patterns Elementary Operations on Matrices The Trace of a Cayley Product that Is Square The Determinant of a Cayley Product of Square Matrices Multiplication of Matrices and Vectors Outer Products Bilinear and Quadratic Forms; Definiteness Anisometric Spaces Other Kinds of Matrix Multiplication Matrix Rank and the Inverse of a Matrix The Rank of Partitioned Matrices, Products of Matrices, and Sums of Matrices Full Rank Partitioning Full Rank Matrices and Matrix Inverses Full Rank Factorization Equivalent Matrices Multiplication by Full Rank Matrices Gramian Matrices: Products of the Form A T A A Lower Bound on the Rank of a Matrix Product Determinants of Inverses Inverses of Products and Sums of Nonsingular Matrices Inverses of Matrices with Special Forms Determining the Rank of a Matrix More on Partitioned Square Matrices: The Schur Complement Inverses of Partitioned Matrices Determinants of Partitioned Matrices Linear Systems of Equations

3 Contents xix Solutions of Linear Systems Null Space: The Orthogonal Complement Generalized Inverses Special Generalized Inverses; The Moore-Penrose Inverse Generalized Inverses of Products and Sums of Matrices Generalized Inverses of Partitioned Matrices Orthogonality Eigenanalysis; Canonical Factorizations Basic Properties of Eigenvalues and Eigenvectors The Characteristic Polynomial The Spectrum Similarity Transformations Schur Factorization Similar Canonical Factorization; Diagonalizable Matrices Properties of Diagonalizable Matrices Eigenanalysis of Symmetric Matrices Positive Definite and Nonnegative Definite Matrices Generalized Eigenvalues and Eigenvectors Singular Values and the Singular Value Decomposition (SVD) Matrix Norms Matrix Norms Induced from Vector Norms The Frobenius Norm The Usual Norm Other Matrix Norms Matrix Norm Inequalities The Spectral Radius Convergence of a Matrix Power Series Approximation of Matrices Exercises Vector/Matrix Derivatives and Integrals Basics of Differentiation Types of Differentiation Differentiation with Respect to a Scalar Differentiation with Respect to a Vector Differentiation with Respect to a Matrix Optimization of Scalar-Valued Functions Stationary Points of Functions Newton s Method Least Squares Maximum Likelihood Optimization of Functions with Constraints Optimization without Differentiation Integration and Expectation: Applications to Probability Distributions Multidimensional Integrals and Integrals InvolvingVectors and Matrices Integration Combined with Other Operations

4 xx Contents Random Variables and Probability Distributions Exercises Matrix Transformations and Factorizations Linear Geometric Transformations Transformations by Orthogonal Matrices Rotations Reflections Translations; Homogeneous Coordinates Householder Transformations (Reflections) Givens Transformations (Rotations) Factorization of Matrices LU and LDU Factorizations QR Factorization Householder Reflections to Form the QR Factorization Givens Rotations to Form the QR Factorization Gram-Schmidt Transformations to Form theqr Factorization Factorizations of Nonnegative Definite Matrices Square Roots Cholesky Factorization Factorizations of a Gramian Matrix Approximate Matrix Factorization Nonnegative Matrix Factorization Incomplete Factorizations Exercises Solution of Linear Systems Condition of Matrices Condition Number Improving the Condition Number Numerical Accuracy Direct Methods for Consistent Systems Gaussian Elimination and Matrix Factorizations Choice of Direct Method Iterative Methods for Consistent Systems The Gauss-Seidel Method withsuccessive Overrelaxation Conjugate Gradient Methods for SymmetricPositive Definite Systems Multigrid Methods Iterative Refinement Updating a Solution to a Consistent System Overdetermined Systems; Least Squares Least Squares Solution of an Overdetermined System Least Squares with a Full Rank Coefficient Matrix Least Squares with a Coefficient MatrixNot of Full Rank Weighted Least Squares

5 Contents xxi Updating a Least Squares Solution of anoverdetermined System Other Solutions of Overdetermined Systems Solutions that Minimize Other Norms of the Residuals Regularized Solutions Minimizing Orthogonal Distances Exercises Evaluation of Eigenvalues and Eigenvectors General Computational Methods Numerical Condition of an Eigenvalue Problem Eigenvalues from Eigenvectors and Vice Versa Deflation Preconditioning Shifting Power Method Jacobi Method QR Method Krylov Methods Generalized Eigenvalues Singular Value Decomposition Exercises Part II Applications in Data Analysis 8 Special Matrices and Operations Useful in Modeling anddata Analysis Data Matrices and Association Matrices Flat Files Graphs and Other Data Structures Term-by-Document Matrices Probability Distribution Models Derived Association Matrices Symmetric Matrices and Other Unitarily Diagonalizable Matrices Some Important Properties of Symmetric Matrices Approximation of Symmetric Matrices and an Important Inequality Normal Matrices Nonnegative Definite Matrices; Cholesky Factorization Positive Definite Matrices Idempotent and Projection Matrices Idempotent Matrices Projection Matrices: Symmetric Idempotent Matrices Special Matrices Occurring in Data Analysis Gramian Matrices Projection and Smoothing Matrices Centered Matrices and Variance-Covariance Matrices.. 363

6 xxii Contents The Generalized Variance Similarity Matrices Dissimilarity Matrices Nonnegative and Positive Matrices Properties of Square Positive Matrices Irreducible Square Nonnegative Matrices Stochastic Matrices Leslie Matrices Other Matrices with Special Structures Helmert Matrices Vandermonde Matrices Hadamard Matrices and Orthogonal Arrays Toeplitz Matrices Circulant Matrices Fourier Matrices and the Discrete Fourier Transform Hankel Matrices Cauchy Matrices Matrices Useful in Graph Theory Z-Matrices and M-Matrices Exercises Selected Applications in Statistics Multivariate Probability Distributions Basic Definitions and Properties The Multivariate Normal Distribution Derived Distributions and Cochran s Theorem Linear Models Fitting the Model Linear Models and Least Squares Statistical Inference The Normal Equations and the Sweep Operator Linear Least Squares Subject to LinearEquality Constraints Weighted Least Squares Updating Linear Regression Statistics Linear Smoothing Multivariate Linear Models Principal Components Principal Components of a Random Vector Principal Components of Data Condition of Models and Data Ill-Conditioning in Statistical Applications Variable Selection Principal Components Regression Shrinkage Estimation Statistical Inference about the Rank of a Matrix

7 Contents xxiii Incomplete Data Optimal Design Multivariate Random Number Generation Stochastic Processes Markov Chains Markovian Population Models Autoregressive Processes Exercises Part III Numerical Methods and Software 10 Numerical Methods Digital Representation of Numeric Data The Fixed-Point Number System The Floating-Point Model for Real Numbers Language Constructs for Representing Numeric Data Other Variations in the Representation of Data;Portability of Data Computer Operations on Numeric Data Fixed-Point Operations Floating-Point Operations Language Constructs for Operations onnumeric Data Software Methods for Extending the Precision Exact Computations Numerical Algorithms and Analysis Error in Numerical Computations Efficiency Iterations and Convergence Other Computational Techniques Exercises Numerical Linear Algebra Computer Storage of Vectors and Matrices General Computational Considerations forvectors and Matrices Relative Magnitudes of Operands Iterative Methods Assessing Computational Errors Multiplication of Vectors and Matrices Other Matrix Computations Exercises Software for Numerical Linear Algebra General Considerations Software Design Software Development, Maintenance, and Testing

8 xxiv Contents Reproducible Research Software Libraries BLAS Level 2 and Level 3 BLAS, LAPACK, and Related Libraries Libraries for High Performance Computing The IMSL Libraries General Purpose Languages Programming Considerations Modern Fortran C and C Python Interactive Systems for Array Manipulation R MATLAB and Octave Exercises Appendices and Back Matter A Notation and Definitions A.1 General Notation A.2 Computer Number Systems A.3 General Mathematical Functions and Operators A.4 Linear Spaces and Matrices A.5 Models and Data B Solutions and Hints for Selected Exercises Bibliography Index

Hands-on Matrix Algebra Using R

Hands-on Matrix Algebra Using R Preface vii 1. R Preliminaries 1 1.1 Matrix Defined, Deeper Understanding Using Software.. 1 1.2 Introduction, Why R?.................... 2 1.3 Obtaining R.......................... 4 1.4 Reference Manuals

More information

Numerical Methods in Matrix Computations

Numerical Methods in Matrix Computations Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices

More information

APPLIED NUMERICAL LINEAR ALGEBRA

APPLIED NUMERICAL LINEAR ALGEBRA APPLIED NUMERICAL LINEAR ALGEBRA James W. Demmel University of California Berkeley, California Society for Industrial and Applied Mathematics Philadelphia Contents Preface 1 Introduction 1 1.1 Basic Notation

More information

Applied Linear Algebra

Applied Linear Algebra Applied Linear Algebra Peter J. Olver School of Mathematics University of Minnesota Minneapolis, MN 55455 olver@math.umn.edu http://www.math.umn.edu/ olver Chehrzad Shakiban Department of Mathematics University

More information

Applied Linear Algebra in Geoscience Using MATLAB

Applied 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 information

Index. book 2009/5/27 page 121. (Page numbers set in bold type indicate the definition of an entry.)

Index. book 2009/5/27 page 121. (Page numbers set in bold type indicate the definition of an entry.) page 121 Index (Page numbers set in bold type indicate the definition of an entry.) A absolute error...26 componentwise...31 in subtraction...27 normwise...31 angle in least squares problem...98,99 approximation

More information

Preface to the Second Edition. Preface to the First Edition

Preface to the Second Edition. Preface to the First Edition n page v Preface to the Second Edition Preface to the First Edition xiii xvii 1 Background in Linear Algebra 1 1.1 Matrices................................. 1 1.2 Square Matrices and Eigenvalues....................

More information

1 Number Systems and Errors 1

1 Number Systems and Errors 1 Contents 1 Number Systems and Errors 1 1.1 Introduction................................ 1 1.2 Number Representation and Base of Numbers............. 1 1.2.1 Normalized Floating-point Representation...........

More information

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 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 information

Introduction to Applied Linear Algebra with MATLAB

Introduction to Applied Linear Algebra with MATLAB Sigam Series in Applied Mathematics Volume 7 Rizwan Butt Introduction to Applied Linear Algebra with MATLAB Heldermann Verlag Contents Number Systems and Errors 1 1.1 Introduction 1 1.2 Number Representation

More information

Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012

Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.

More information

Index. for generalized eigenvalue problem, butterfly form, 211

Index. for generalized eigenvalue problem, butterfly form, 211 Index ad hoc shifts, 165 aggressive early deflation, 205 207 algebraic multiplicity, 35 algebraic Riccati equation, 100 Arnoldi process, 372 block, 418 Hamiltonian skew symmetric, 420 implicitly restarted,

More information

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2 Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition

More information

Index. C (programming language), 387, 401,

Index. C (programming language), 387, 401, Index A-optimality, 356 absolute error, 395, 404, 434 ACM Transactions on Mathematical Software, 505 adj( ), 53 adjacency matrix, 265, 266, 314 adjoint (see also conjugate transpose), 44 adjoint, classical

More information

Introduction to Numerical Analysis

Introduction to Numerical Analysis J. Stoer R. Bulirsch Introduction to Numerical Analysis Second Edition Translated by R. Bartels, W. Gautschi, and C. Witzgall With 35 Illustrations Springer Contents Preface to the Second Edition Preface

More information

Linear Algebra Massoud Malek

Linear 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 information

MATRIX AND LINEAR ALGEBR A Aided with MATLAB

MATRIX AND LINEAR ALGEBR A Aided with MATLAB Second Edition (Revised) MATRIX AND LINEAR ALGEBR A Aided with MATLAB Kanti Bhushan Datta Matrix and Linear Algebra Aided with MATLAB Second Edition KANTI BHUSHAN DATTA Former Professor Department of Electrical

More information

NUMERICAL COMPUTATION IN SCIENCE AND ENGINEERING

NUMERICAL COMPUTATION IN SCIENCE AND ENGINEERING NUMERICAL COMPUTATION IN SCIENCE AND ENGINEERING C. Pozrikidis University of California, San Diego New York Oxford OXFORD UNIVERSITY PRESS 1998 CONTENTS Preface ix Pseudocode Language Commands xi 1 Numerical

More information

MATHEMATICS. Course Syllabus. Section A: Linear Algebra. Subject Code: MA. Course Structure. Ordinary Differential Equations

MATHEMATICS. Course Syllabus. Section A: Linear Algebra. Subject Code: MA. Course Structure. Ordinary Differential Equations MATHEMATICS Subject Code: MA Course Structure Sections/Units Section A Section B Section C Linear Algebra Complex Analysis Real Analysis Topics Section D Section E Section F Section G Section H Section

More information

FINITE-DIMENSIONAL LINEAR ALGEBRA

FINITE-DIMENSIONAL LINEAR ALGEBRA DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H ROSEN FINITE-DIMENSIONAL LINEAR ALGEBRA Mark S Gockenbach Michigan Technological University Houghton, USA CRC Press Taylor & Francis Croup

More information

Properties of Matrices and Operations on Matrices

Properties 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 information

Index. Copyright (c)2007 The Society for Industrial and Applied Mathematics From: Matrix Methods in Data Mining and Pattern Recgonition By: Lars Elden

Index. Copyright (c)2007 The Society for Industrial and Applied Mathematics From: Matrix Methods in Data Mining and Pattern Recgonition By: Lars Elden Index 1-norm, 15 matrix, 17 vector, 15 2-norm, 15, 59 matrix, 17 vector, 15 3-mode array, 91 absolute error, 15 adjacency matrix, 158 Aitken extrapolation, 157 algebra, multi-linear, 91 all-orthogonality,

More information

A THEORETICAL INTRODUCTION TO NUMERICAL ANALYSIS

A THEORETICAL INTRODUCTION TO NUMERICAL ANALYSIS A THEORETICAL INTRODUCTION TO NUMERICAL ANALYSIS Victor S. Ryaben'kii Semyon V. Tsynkov Chapman &. Hall/CRC Taylor & Francis Group Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor

More information

Math 307 Learning Goals. March 23, 2010

Math 307 Learning Goals. March 23, 2010 Math 307 Learning Goals March 23, 2010 Course Description The course presents core concepts of linear algebra by focusing on applications in Science and Engineering. Examples of applications from recent

More information

Statistical Signal Processing Detection, Estimation, and Time Series Analysis

Statistical Signal Processing Detection, Estimation, and Time Series Analysis Statistical Signal Processing Detection, Estimation, and Time Series Analysis Louis L. Scharf University of Colorado at Boulder with Cedric Demeure collaborating on Chapters 10 and 11 A TT ADDISON-WESLEY

More information

LinGloss. A glossary of linear algebra

LinGloss. 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 information

Numerical Methods - Numerical Linear Algebra

Numerical Methods - Numerical Linear Algebra Numerical Methods - Numerical Linear Algebra Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Numerical Linear Algebra I 2013 1 / 62 Outline 1 Motivation 2 Solving Linear

More information

Contents. Preface... xi. Introduction...

Contents. Preface... xi. Introduction... Contents Preface... xi Introduction... xv Chapter 1. Computer Architectures... 1 1.1. Different types of parallelism... 1 1.1.1. Overlap, concurrency and parallelism... 1 1.1.2. Temporal and spatial parallelism

More information

Contents. Preface to the Third Edition (2007) Preface to the Second Edition (1992) Preface to the First Edition (1985) License and Legal Information

Contents. Preface to the Third Edition (2007) Preface to the Second Edition (1992) Preface to the First Edition (1985) License and Legal Information Contents Preface to the Third Edition (2007) Preface to the Second Edition (1992) Preface to the First Edition (1985) License and Legal Information xi xiv xvii xix 1 Preliminaries 1 1.0 Introduction.............................

More information

MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT

MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT The following is the list of questions for the oral exam. At the same time, these questions represent all topics for the written exam. The procedure for

More information

Applied Linear Algebra in Geoscience Using MATLAB

Applied 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 information

Linear Algebra and Probability

Linear Algebra and Probability Linear Algebra and Probability for Computer Science Applications Ernest Davis CRC Press Taylor!* Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor Sc Francis Croup, an informa

More information

Numerical Methods. Elena loli Piccolomini. Civil Engeneering. piccolom. Metodi Numerici M p. 1/??

Numerical 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 information

6.4 Krylov Subspaces and Conjugate Gradients

6.4 Krylov Subspaces and Conjugate Gradients 6.4 Krylov Subspaces and Conjugate Gradients Our original equation is Ax = b. The preconditioned equation is P Ax = P b. When we write P, we never intend that an inverse will be explicitly computed. P

More information

ITERATIVE METHODS FOR SPARSE LINEAR SYSTEMS

ITERATIVE METHODS FOR SPARSE LINEAR SYSTEMS ITERATIVE METHODS FOR SPARSE LINEAR SYSTEMS YOUSEF SAAD University of Minnesota PWS PUBLISHING COMPANY I(T)P An International Thomson Publishing Company BOSTON ALBANY BONN CINCINNATI DETROIT LONDON MADRID

More information

1 Cricket chirps: an example

1 Cricket chirps: an example Notes for 2016-09-26 1 Cricket chirps: an example Did you know that you can estimate the temperature by listening to the rate of chirps? The data set in Table 1 1. represents measurements of the number

More information

Review problems for MA 54, Fall 2004.

Review problems for MA 54, Fall 2004. Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on

More information

Statistics for Social and Behavioral Sciences

Statistics for Social and Behavioral Sciences Statistics for Social and Behavioral Sciences Advisors: S.E. Fienberg W.J. van der Linden For other titles published in this series, go to http://www.springer.com/series/3463 Haruo Yanai Kei Takeuchi

More information

MAT 610: Numerical Linear Algebra. James V. Lambers

MAT 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 information

homogeneous 71 hyperplane 10 hyperplane 34 hyperplane 69 identity map 171 identity map 186 identity map 206 identity matrix 110 identity matrix 45

homogeneous 71 hyperplane 10 hyperplane 34 hyperplane 69 identity map 171 identity map 186 identity map 206 identity matrix 110 identity matrix 45 address 12 adjoint matrix 118 alternating 112 alternating 203 angle 159 angle 33 angle 60 area 120 associative 180 augmented matrix 11 axes 5 Axiom of Choice 153 basis 178 basis 210 basis 74 basis test

More information

Adaptive Filtering. Squares. Alexander D. Poularikas. Fundamentals of. Least Mean. with MATLABR. University of Alabama, Huntsville, AL.

Adaptive Filtering. Squares. Alexander D. Poularikas. Fundamentals of. Least Mean. with MATLABR. University of Alabama, Huntsville, AL. Adaptive Filtering Fundamentals of Least Mean Squares with MATLABR Alexander D. Poularikas University of Alabama, Huntsville, AL CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is

More information

Principal Component Analysis

Principal Component Analysis I.T. Jolliffe Principal Component Analysis Second Edition With 28 Illustrations Springer Contents Preface to the Second Edition Preface to the First Edition Acknowledgments List of Figures List of Tables

More information

Math 307 Learning Goals

Math 307 Learning Goals Math 307 Learning Goals May 14, 2018 Chapter 1 Linear Equations 1.1 Solving Linear Equations Write a system of linear equations using matrix notation. Use Gaussian elimination to bring a system of linear

More information

Chapter 7 Iterative Techniques in Matrix Algebra

Chapter 7 Iterative Techniques in Matrix Algebra Chapter 7 Iterative Techniques in Matrix Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128B Numerical Analysis Vector Norms Definition

More information

Review of some mathematical tools

Review 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 information

Coding the Matrix Index - Version 0

Coding the Matrix Index - Version 0 0 vector, [definition]; (2.4.1): 68 2D geometry, transformations in, [lab]; (4.15.0): 196-200 A T (matrix A transpose); (4.5.4): 157 absolute value, complex number; (1.4.1): 43 abstract/abstracting, over

More information

ABSTRACT ALGEBRA WITH APPLICATIONS

ABSTRACT ALGEBRA WITH APPLICATIONS ABSTRACT ALGEBRA WITH APPLICATIONS IN TWO VOLUMES VOLUME I VECTOR SPACES AND GROUPS KARLHEINZ SPINDLER Darmstadt, Germany Marcel Dekker, Inc. New York Basel Hong Kong Contents f Volume I Preface v VECTOR

More information

Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Dennis S. Bernstein

Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Dennis S. Bernstein Matrix Mathematics Theory, Facts, and Formulas with Application to Linear Systems Theory Dennis S. Bernstein PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Contents Special Symbols xv Conventions, Notation,

More information

Mobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti

Mobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti Mobile Robotics 1 A Compact Course on Linear Algebra Giorgio Grisetti SA-1 Vectors Arrays of numbers They represent a point in a n dimensional space 2 Vectors: Scalar Product Scalar-Vector Product Changes

More information

Math 302 Outcome Statements Winter 2013

Math 302 Outcome Statements Winter 2013 Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a

More information

The Conjugate Gradient Method

The Conjugate Gradient Method The Conjugate Gradient Method Classical Iterations We have a problem, We assume that the matrix comes from a discretization of a PDE. The best and most popular model problem is, The matrix will be as large

More information

Numerical Methods with MATLAB

Numerical Methods with MATLAB Numerical Methods with MATLAB A Resource for Scientists and Engineers G. J. BÖRSE Lehigh University PWS Publishing Company I(T)P AN!NTERNATIONAL THOMSON PUBLISHING COMPANY Boston Albany Bonn Cincinnati

More information

LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12,

LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12, LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12, 2000 74 6 Summary Here we summarize the most important information about theoretical and numerical linear algebra. MORALS OF THE STORY: I. Theoretically

More information

Numerical Mathematics

Numerical Mathematics Alfio Quarteroni Riccardo Sacco Fausto Saleri Numerical Mathematics Second Edition With 135 Figures and 45 Tables 421 Springer Contents Part I Getting Started 1 Foundations of Matrix Analysis 3 1.1 Vector

More information

ADAPTIVE FILTER THEORY

ADAPTIVE FILTER THEORY ADAPTIVE FILTER THEORY Fourth Edition Simon Haykin Communications Research Laboratory McMaster University Hamilton, Ontario, Canada Front ice Hall PRENTICE HALL Upper Saddle River, New Jersey 07458 Preface

More information

Lecture 7: Positive Semidefinite Matrices

Lecture 7: Positive Semidefinite Matrices Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.

More information

Algebra C Numerical Linear Algebra Sample Exam Problems

Algebra C Numerical Linear Algebra Sample Exam Problems Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric

More information

Notes on Eigenvalues, Singular Values and QR

Notes 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 information

MATCOM and Selected MATCOM Functions

MATCOM and Selected MATCOM Functions book-onlin page 567 Appendix C MATCOM and Selected MATCOM Functions C.1 MATCOM and Selected MATCOM Functions C.1.1 What is MATCOM? MATCOM is a MATLAB-based interactive software package containing the implementation

More information

Linear Models 1. Isfahan University of Technology Fall Semester, 2014

Linear Models 1. Isfahan University of Technology Fall Semester, 2014 Linear Models 1 Isfahan University of Technology Fall Semester, 2014 References: [1] G. A. F., Seber and A. J. Lee (2003). Linear Regression Analysis (2nd ed.). Hoboken, NJ: Wiley. [2] A. C. Rencher and

More information

Albert W. Marshall. Ingram Olkin Barry. C. Arnold. Inequalities: Theory. of Majorization and Its Applications. Second Edition.

Albert W. Marshall. Ingram Olkin Barry. C. Arnold. Inequalities: Theory. of Majorization and Its Applications. Second Edition. Albert W Marshall Ingram Olkin Barry C Arnold Inequalities: Theory of Majorization and Its Applications Second Edition f) Springer Contents I Theory of Majorization 1 Introduction 3 A Motivation and Basic

More information

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N.

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N. Math 410 Homework Problems In the following pages you will find all of the homework problems for the semester. Homework should be written out neatly and stapled and turned in at the beginning of class

More information

Wiley. Methods and Applications of Linear Models. Regression and the Analysis. of Variance. Third Edition. Ishpeming, Michigan RONALD R.

Wiley. Methods and Applications of Linear Models. Regression and the Analysis. of Variance. Third Edition. Ishpeming, Michigan RONALD R. Methods and Applications of Linear Models Regression and the Analysis of Variance Third Edition RONALD R. HOCKING PenHock Statistical Consultants Ishpeming, Michigan Wiley Contents Preface to the Third

More information

Matrix Differential Calculus with Applications in Statistics and Econometrics

Matrix Differential Calculus with Applications in Statistics and Econometrics Matrix Differential Calculus with Applications in Statistics and Econometrics Revised Edition JAN. R. MAGNUS CentERjor Economic Research, Tilburg University and HEINZ NEUDECKER Cesaro, Schagen JOHN WILEY

More information

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,

More information

Matrix Algorithms. Volume II: Eigensystems. G. W. Stewart H1HJ1L. University of Maryland College Park, Maryland

Matrix Algorithms. Volume II: Eigensystems. G. W. Stewart H1HJ1L. University of Maryland College Park, Maryland Matrix Algorithms Volume II: Eigensystems G. W. Stewart University of Maryland College Park, Maryland H1HJ1L Society for Industrial and Applied Mathematics Philadelphia CONTENTS Algorithms Preface xv xvii

More information

GEOPHYSICAL INVERSE THEORY AND REGULARIZATION PROBLEMS

GEOPHYSICAL INVERSE THEORY AND REGULARIZATION PROBLEMS Methods in Geochemistry and Geophysics, 36 GEOPHYSICAL INVERSE THEORY AND REGULARIZATION PROBLEMS Michael S. ZHDANOV University of Utah Salt Lake City UTAH, U.S.A. 2OO2 ELSEVIER Amsterdam - Boston - London

More information

Seminar on Linear Algebra

Seminar on Linear Algebra Supplement Seminar on Linear Algebra Projection, Singular Value Decomposition, Pseudoinverse Kenichi Kanatani Kyoritsu Shuppan Co., Ltd. Contents 1 Linear Space and Projection 1 1.1 Expression of Linear

More information

Lecture 2: Linear Algebra Review

Lecture 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 information

Statistical Geometry Processing Winter Semester 2011/2012

Statistical Geometry Processing Winter Semester 2011/2012 Statistical Geometry Processing Winter Semester 2011/2012 Linear Algebra, Function Spaces & Inverse Problems Vector and Function Spaces 3 Vectors vectors are arrows in space classically: 2 or 3 dim. Euclidian

More information

Linear Algebra Review

Linear Algebra Review Linear Algebra Review CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Linear Algebra Review 1 / 16 Midterm Exam Tuesday Feb

More information

Mathematical Methods for Engineers and Scientists 1

Mathematical Methods for Engineers and Scientists 1 K.T. Tang Mathematical Methods for Engineers and Scientists 1 Complex Analysis, Determinants and Matrices With 49 Figures and 2 Tables fyj Springer Part I Complex Analysis 1 Complex Numbers 3 1.1 Our Number

More information

Geometric Modeling Summer Semester 2010 Mathematical Tools (1)

Geometric Modeling Summer Semester 2010 Mathematical Tools (1) Geometric Modeling Summer Semester 2010 Mathematical Tools (1) Recap: Linear Algebra Today... Topics: Mathematical Background Linear algebra Analysis & differential geometry Numerical techniques Geometric

More information

NUMERICAL MATHEMATICS AND COMPUTING

NUMERICAL MATHEMATICS AND COMPUTING NUMERICAL MATHEMATICS AND COMPUTING Fourth Edition Ward Cheney David Kincaid The University of Texas at Austin 9 Brooks/Cole Publishing Company I(T)P An International Thomson Publishing Company Pacific

More information

Lecture 2: Numerical linear algebra

Lecture 2: Numerical linear algebra Lecture 2: Numerical linear algebra QR factorization Eigenvalue decomposition Singular value decomposition Conditioning of a problem Floating point arithmetic and stability of an algorithm Linear algebra

More information

ALG - Algebra

ALG - Algebra Coordinating unit: Teaching unit: Academic year: Degree: ECTS credits: 2018 270 - FIB - Barcelona School of Informatics 749 - MAT - Department of Mathematics BACHELOR'S DEGREE IN DATA SCIENCE AND ENGINEERING

More information

Multivariate Statistical Analysis

Multivariate Statistical Analysis Multivariate Statistical Analysis Fall 2011 C. L. Williams, Ph.D. Lecture 4 for Applied Multivariate Analysis Outline 1 Eigen values and eigen vectors Characteristic equation Some properties of eigendecompositions

More information

7. Symmetric Matrices and Quadratic Forms

7. 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 information

An Introduction to Multivariate Statistical Analysis

An Introduction to Multivariate Statistical Analysis An Introduction to Multivariate Statistical Analysis Third Edition T. W. ANDERSON Stanford University Department of Statistics Stanford, CA WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Contents

More information

NUMERICAL METHODS FOR ENGINEERING APPLICATION

NUMERICAL METHODS FOR ENGINEERING APPLICATION NUMERICAL METHODS FOR ENGINEERING APPLICATION Second Edition JOEL H. FERZIGER A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto

More information

(Mathematical Operations with Arrays) Applied Linear Algebra in Geoscience Using MATLAB

(Mathematical Operations with Arrays) Applied Linear Algebra in Geoscience Using MATLAB Applied Linear Algebra in Geoscience Using MATLAB (Mathematical Operations with Arrays) Contents Getting Started Matrices Creating Arrays Linear equations Mathematical Operations with Arrays Using Script

More information

Chapter 3 Transformations

Chapter 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 information

Time Series: Theory and Methods

Time Series: Theory and Methods Peter J. Brockwell Richard A. Davis Time Series: Theory and Methods Second Edition With 124 Illustrations Springer Contents Preface to the Second Edition Preface to the First Edition vn ix CHAPTER 1 Stationary

More information

Introduction to Numerical Linear Algebra II

Introduction to Numerical Linear Algebra II Introduction to Numerical Linear Algebra II Petros Drineas These slides were prepared by Ilse Ipsen for the 2015 Gene Golub SIAM Summer School on RandNLA 1 / 49 Overview We will cover this material in

More information

GATE Engineering Mathematics SAMPLE STUDY MATERIAL. Postal Correspondence Course GATE. Engineering. Mathematics GATE ENGINEERING MATHEMATICS

GATE Engineering Mathematics SAMPLE STUDY MATERIAL. Postal Correspondence Course GATE. Engineering. Mathematics GATE ENGINEERING MATHEMATICS SAMPLE STUDY MATERIAL Postal Correspondence Course GATE Engineering Mathematics GATE ENGINEERING MATHEMATICS ENGINEERING MATHEMATICS GATE Syllabus CIVIL ENGINEERING CE CHEMICAL ENGINEERING CH MECHANICAL

More information

Linear Algebra in Actuarial Science: Slides to the lecture

Linear 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 information

Matrix Factorizations

Matrix Factorizations 1 Stat 540, Matrix Factorizations Matrix Factorizations LU Factorization Definition... Given a square k k matrix S, the LU factorization (or decomposition) represents S as the product of two triangular

More information

MATH36001 Generalized Inverses and the SVD 2015

MATH36001 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 information

Chap 3. Linear Algebra

Chap 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 information

ADAPTIVE FILTER THEORY

ADAPTIVE FILTER THEORY ADAPTIVE FILTER THEORY Fifth Edition Simon Haykin Communications Research Laboratory McMaster University Hamilton, Ontario, Canada International Edition contributions by Telagarapu Prabhakar Department

More information

Syllabus for Applied Mathematics Graduate Student Qualifying Exams, Dartmouth Mathematics Department

Syllabus for Applied Mathematics Graduate Student Qualifying Exams, Dartmouth Mathematics Department Syllabus for Applied Mathematics Graduate Student Qualifying Exams, Dartmouth Mathematics Department Alex Barnett, Scott Pauls, Dan Rockmore August 12, 2011 We aim to touch upon many topics that a professional

More information

Statistical and Adaptive Signal Processing

Statistical and Adaptive Signal Processing r Statistical and Adaptive Signal Processing Spectral Estimation, Signal Modeling, Adaptive Filtering and Array Processing Dimitris G. Manolakis Massachusetts Institute of Technology Lincoln Laboratory

More information

Course Notes: Week 1

Course Notes: Week 1 Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra 1 Lecture 1: Introduction (3/28/11) We will focus on iterative methods for solving linear systems of equations (and some discussion of eigenvalues

More information

Introduction. Chapter One

Introduction. Chapter One Chapter One Introduction The aim of this book is to describe and explain the beautiful mathematical relationships between matrices, moments, orthogonal polynomials, quadrature rules and the Lanczos and

More information

CHAPTER 11. A Revision. 1. The Computers and Numbers therein

CHAPTER 11. A Revision. 1. The Computers and Numbers therein CHAPTER A Revision. The Computers and Numbers therein Traditional computer science begins with a finite alphabet. By stringing elements of the alphabet one after another, one obtains strings. A set of

More information

1 Vectors. Notes for Bindel, Spring 2017 Numerical Analysis (CS 4220)

1 Vectors. Notes for Bindel, Spring 2017 Numerical Analysis (CS 4220) Notes for 2017-01-30 Most of mathematics is best learned by doing. Linear algebra is no exception. You have had a previous class in which you learned the basics of linear algebra, and you will have plenty

More information

BASIC MATRIX ALGEBRA WITH ALGORITHMS AND APPLICATIONS ROBERT A. LIEBLER CHAPMAN & HALL/CRC

BASIC MATRIX ALGEBRA WITH ALGORITHMS AND APPLICATIONS ROBERT A. LIEBLER CHAPMAN & HALL/CRC BASIC MATRIX ALGEBRA WITH ALGORITHMS AND APPLICATIONS ROBERT A. LIEBLER CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. Contents Preface Examples Major results/proofs

More information

MA3025 Course Prerequisites

MA3025 Course Prerequisites MA3025 Course Prerequisites MA 3025 (4-1) MA3025 (4-1) Logic and Discrete Mathematics: Provides a rigorous foundation in logic and elementary discrete mathematics. Topics from logic include modeling English

More information

BASIC EXAM ADVANCED CALCULUS/LINEAR ALGEBRA

BASIC EXAM ADVANCED CALCULUS/LINEAR ALGEBRA 1 BASIC EXAM ADVANCED CALCULUS/LINEAR ALGEBRA This part of the Basic Exam covers topics at the undergraduate level, most of which might be encountered in courses here such as Math 233, 235, 425, 523, 545.

More information

1 9/5 Matrices, vectors, and their applications

1 9/5 Matrices, vectors, and their applications 1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric

More information