Linear Algebra & Geometry why is linear algebra useful in computer vision?

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Linear Algebra & Geometry why is linear algebra useful in computer vision?"

Transcription

1 Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia I. Camps, Penn State University

2 Why is linear algebra useful in computer vision? Representation 3D points in the scene 2D points in the image Coordinates will be used to Perform geometrical transformations Associate 3D with 2D points Images are matrices of numbers Find properties of these numbers

3 Agenda 1. Basics definitions and properties 2. Geometrical transformations 3. SVD and its applications

4 Vectors (i.e., 2D or 3D vectors) P = [x,y,z] p = [x,y] 3D world Image

5 Vectors (i.e., 2D vectors) x2 v θ P Magnitude: x1 If, Is a UNIT vector Is a unit vector Orientation:

6 Vector Addition v+w v w

7 Vector Subtraction v v-w w

8 Scalar Product v av

9 Inner (dot) Product v α w The inner product is a SCALAR!

10 Orthonormal Basis x2 j i v θ P x1

11 Vector (cross) Product u w α v The cross product is a VECTOR! Magnitude:kuk = kv wk = kvkkwk sin Orientation:

12 Vector Product Computation i = (1,0,0) j = (0,1,0) k = (0,0,1) i = 1 j = 1 k = 1 i = j k j = k i k = i j = ( x2 y 3 x3 y ) i + 2 ( x3 y 1 x1 y3) j+ ( x1 y 2 x2 y1 ) k

13 Matrices A n m = 2 3 a 11 a a 1m a 21 a a 2m a n1 a n2... a nm Pixel s intensity value Sum: A and B must have the same dimensions! Example:

14 Matrices A n m = 2 3 a 11 a a 1m a 21 a a 2m a n1 a n2... a nm Product: A and B must have compatible dimensions!

15 Matrix Transpose Definition: C m n = A T n m c ij = a ji Identities: (A + B) T = A T + B T (AB) T = B T A T If A = A T, then A is symmetric

16 Matrix Determinant Useful value computed from the elements of a square matrix A det [ a 11 ] = a11 [ ] a11 a det 12 = a a 21 a 11 a 22 a 12 a a 11 a 12 a 13 det a 21 a 22 a 23 = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 31 a 32 a 33 a 13 a 22 a 31 a 23 a 32 a 11 a 33 a 12 a 21

17 Matrix Inverse Does not exist for all matrices, necessary (but not sufficient) that the matrix is square AA 1 = A 1 A = I [ ] 1 A 1 a11 a = 12 = 1 a 21 a 22 det A If det A = 0, A does not have an inverse. [ a22 a 12 a 21 a 11 ], det A 0

18 2D Geometrical Transformations

19 2D Translation P P t

20 2D Translation Equation ty y P t P x tx

21 2D Translation using Matrices ty P t P y x tx

22 Homogeneous Coordinates Multiply the coordinates by a non-zero scalar and add an extra coordinate equal to that scalar. For example,

23 Back to Cartesian Coordinates: Divide by the last coordinate and eliminate it. For example, NOTE: in our example the scalar was 1

24 2D Translation using Homogeneous Coordinates ty P t P y t P x tx

25 Scaling P P

26 Scaling Equation s y y P y P x s x x

27 Scaling & Translating P P P =S P P =T P P =T P =T (S P)=(T S) P = A P

28 Scaling & Translating A

29 Translating & Scaling = Scaling & Translating?

30 Rotation P P

31 Rotation Equations Counter-clockwise rotation by an angle θ y y P x θ x P

32 Degrees of Freedom R is 2x2 4 elements Note: R belongs to the category of normal matrices and satisfies many interesting properties:

33 Rotation+ Scaling +Translation P = (T R S) P If s x =s y, this is a similarity transformation!

34 Transformation in 2D -Isometries -Similarities -Affinity -Projective

35 Transformation in 2D Isometries: [Euclideans] - Preserve distance (areas) - 3 DOF - Regulate motion of rigid object

36 Transformation in 2D Similarities: - Preserve - ratio of lengths - angles -4 DOF

37 Affinities: Transformation in 2D

38 Transformation in 2D Affinities: -Preserve: - Parallel lines - Ratio of areas - Ratio of lengths on collinear lines - others - 6 DOF

39 Transformation in 2D Projective: - 8 DOF - Preserve: - cross ratio of 4 collinear points - collinearity - and a few others

40 Eigenvalues and Eigenvectors A eigenvalue λ and eigenvector u satisfies where A is a square matrix. Au = λu Multiplying u by A scales u by λ

41 Eigenvalues and Eigenvectors Rearranging the previous equation gives the system Au λu = (A λi)u = 0 which has a solution if and only if det(a λi) = 0. The eigenvalues are the roots of this determinant which is polynomial in λ. Substitute the resulting eigenvalues back into Au = λu and solve to obtain the corresponding eigenvector.

42 Singular Value Decomposition UΣV T = A Where U and V are orthogonal matrices, and Σ is a diagonal matrix. For example:

43 Singular Value decomposition that

44 An Numerical Example Look at how the multiplication works out, left to right: Column 1 of U gets scaled by the first value from Σ. The resulting vector gets scaled by row 1 of V T to produce a contribution to the columns of A

45 An Numerical Example + = Each product of (column i of U) (value i from Σ) (row i of V T ) produces a

46 An Numerical Example We can look at Σ to see that the first column has a large effect while the second column has a much smaller effect in this example

47 SVD Applications For this image, using only the first 10 of 300 singular values produces a recognizable reconstruction So, SVD can be used for image compression

48 Principal Component Analysis Remember, columns of U are the Principal Components of the data: the major patterns that can be added to produce the columns of the original matrix One use of this is to construct a matrix where each column is a separate data sample Run SVD on that matrix, and look at the first few columns of U to see patterns that are common among the columns This is called Principal Component Analysis (or PCA) of the data samples

49 Principal Component Analysis Often, raw data samples have a lot of redundancy and patterns PCA can allow you to represent data samples as weights on the principal components, rather than using the original raw form of the data By representing each sample as just those weights, you can represent just the meat of what s different between samples. This minimal representation makes machine learning and other algorithms much more efficient

50

51

Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

More information

Lecture: Face Recognition and Feature Reduction

Lecture: 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 information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear 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 information

Image Registration Lecture 2: Vectors and Matrices

Image 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 information

Singular Value Decomposition

Singular 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 information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Dr Gerhard Roth COMP 40A Winter 05 Version Linear algebra Is an important area of mathematics It is the basis of computer vision Is very widely taught, and there are many resources

More information

1 Last time: least-squares problems

1 Last time: least-squares problems MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that

More information

and let s calculate the image of some vectors under the transformation T.

and let s calculate the image of some vectors under the transformation T. Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

More information

Linear Algebra for Machine Learning. Sargur N. Srihari

Linear Algebra for Machine Learning. Sargur N. Srihari Linear Algebra for Machine Learning Sargur N. srihari@cedar.buffalo.edu 1 Overview Linear Algebra is based on continuous math rather than discrete math Computer scientists have little experience with it

More information

A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].

A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3]. Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes

More information

The Singular Value Decomposition (SVD) and Principal Component Analysis (PCA)

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 information

Background Mathematics (2/2) 1. David Barber

Background 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 information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

More information

Maths for Signals and Systems Linear Algebra in Engineering

Maths 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 information

Math 215 HW #9 Solutions

Math 215 HW #9 Solutions Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith

More information

Linear Algebra V = T = ( 4 3 ).

Linear Algebra V = T = ( 4 3 ). Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional

More 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

Positive Definite Matrix

Positive 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 information

Example Linear Algebra Competency Test

Example Linear Algebra Competency Test Example Linear Algebra Competency Test The 4 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter case,

More information

Lecture 10: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11)

Lecture 10: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11) Lecture 1: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11) The eigenvalue problem, Ax= λ x, occurs in many, many contexts: classical mechanics, quantum mechanics, optics 22 Eigenvectors and

More information

Lecture II: Linear Algebra Revisited

Lecture II: Linear Algebra Revisited Lecture II: Linear Algebra Revisited Overview Vector spaces, Hilbert & Banach Spaces, etrics & Norms atrices, Eigenvalues, Orthogonal Transformations, Singular Values Operators, Operator Norms, Function

More information

Linear Algebra Primer

Linear 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 information

Lecture 3: Matrix and Matrix Operations

Lecture 3: Matrix and Matrix Operations Lecture 3: Matrix and Matrix Operations Representation, row vector, column vector, element of a matrix. Examples of matrix representations Tables and spreadsheets Scalar-Matrix operation: Scaling a matrix

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 =

(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 information

Extra Problems for Math 2050 Linear Algebra I

Extra Problems for Math 2050 Linear Algebra I Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as

More information

DATA MINING LECTURE 8. Dimensionality Reduction PCA -- SVD

DATA MINING LECTURE 8. Dimensionality Reduction PCA -- SVD DATA MINING LECTURE 8 Dimensionality Reduction PCA -- SVD The curse of dimensionality Real data usually have thousands, or millions of dimensions E.g., web documents, where the dimensionality is the vocabulary

More information

Computational math: Assignment 1

Computational 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 information

Fundamentals of Matrices

Fundamentals of Matrices Maschinelles Lernen II Fundamentals of Matrices Christoph Sawade/Niels Landwehr/Blaine Nelson Tobias Scheffer Matrix Examples Recap: Data Linear Model: f i x = w i T x Let X = x x n be the data matrix

More information

Linear Algebra and Eigenproblems

Linear Algebra and Eigenproblems Appendix A A Linear Algebra and Eigenproblems A working knowledge of linear algebra is key to understanding many of the issues raised in this work. In particular, many of the discussions of the details

More information

Quiz ) Locate your 1 st order neighbors. 1) Simplify. Name Hometown. Name Hometown. Name Hometown.

Quiz ) Locate your 1 st order neighbors. 1) Simplify. Name Hometown. Name Hometown. Name Hometown. Quiz 1) Simplify 9999 999 9999 998 9999 998 2) Locate your 1 st order neighbors Name Hometown Me Name Hometown Name Hometown Name Hometown Solving Linear Algebraic Equa3ons Basic Concepts Here only real

More information

1. General Vector Spaces

1. 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 information

Lecture 6 Positive Definite Matrices

Lecture 6 Positive Definite Matrices Linear Algebra Lecture 6 Positive Definite Matrices Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices

More information

Vectors and Matrices Statistics with Vectors and Matrices

Vectors and Matrices Statistics with Vectors and Matrices Vectors and Matrices Statistics with Vectors and Matrices Lecture 3 September 7, 005 Analysis Lecture #3-9/7/005 Slide 1 of 55 Today s Lecture Vectors and Matrices (Supplement A - augmented with SAS proc

More information

Functional Analysis Review

Functional 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 information

COMP 558 lecture 18 Nov. 15, 2010

COMP 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 information

Linear Algebra and Matrices

Linear Algebra and Matrices Linear Algebra and Matrices 4 Overview In this chapter we studying true matrix operations, not element operations as was done in earlier chapters. Working with MAT- LAB functions should now be fairly routine.

More information

Computational Methods. Eigenvalues and Singular Values

Computational 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 information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

Matrices and Deformation

Matrices and Deformation ES 111 Mathematical Methods in the Earth Sciences Matrices and Deformation Lecture Outline 13 - Thurs 9th Nov 2017 Strain Ellipse and Eigenvectors One way of thinking about a matrix is that it operates

More information

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)

MATH 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 information

Matrices and Matrix Algebra.

Matrices and Matrix Algebra. Matrices and Matrix Algebra 3.1. Operations on Matrices Matrix Notation and Terminology Matrix: a rectangular array of numbers, called entries. A matrix with m rows and n columns m n A n n matrix : a square

More information

2. Review of Linear Algebra

2. 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 information

AM 205: lecture 8. Last time: Cholesky factorization, QR factorization Today: how to compute the QR factorization, the Singular Value Decomposition

AM 205: lecture 8. Last time: Cholesky factorization, QR factorization Today: how to compute the QR factorization, the Singular Value Decomposition AM 205: lecture 8 Last time: Cholesky factorization, QR factorization Today: how to compute the QR factorization, the Singular Value Decomposition QR Factorization A matrix A R m n, m n, can be factorized

More information

EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants

EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For

More information

ANSWERS. E k E 2 E 1 A = B

ANSWERS. E k E 2 E 1 A = B MATH 7- Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,

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

Stat 159/259: Linear Algebra Notes

Stat 159/259: Linear Algebra Notes Stat 159/259: Linear Algebra Notes Jarrod Millman November 16, 2015 Abstract These notes assume you ve taken a semester of undergraduate linear algebra. In particular, I assume you are familiar with the

More information

Applied Mathematics 205. Unit II: Numerical Linear Algebra. Lecturer: Dr. David Knezevic

Applied 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 information

Face Recognition and Biometric Systems

Face Recognition and Biometric Systems The Eigenfaces method Plan of the lecture Principal Components Analysis main idea Feature extraction by PCA face recognition Eigenfaces training feature extraction Literature M.A.Turk, A.P.Pentland Face

More information

Vector Spaces, Affine Spaces, and Metric Spaces

Vector Spaces, Affine Spaces, and Metric Spaces Vector Spaces, Affine Spaces, and Metric Spaces 2 This chapter is only meant to give a short overview of the most important concepts in linear algebra, affine spaces, and metric spaces and is not intended

More information

18.06SC Final Exam Solutions

18.06SC Final Exam Solutions 18.06SC Final Exam Solutions 1 (4+7=11 pts.) Suppose A is 3 by 4, and Ax = 0 has exactly 2 special solutions: 1 2 x 1 = 1 and x 2 = 1 1 0 0 1 (a) Remembering that A is 3 by 4, find its row reduced echelon

More information

Principal Component Analysis

Principal Component Analysis Principal Component Analysis CS5240 Theoretical Foundations in Multimedia Leow Wee Kheng Department of Computer Science School of Computing National University of Singapore Leow Wee Kheng (NUS) Principal

More information

Review of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem

Review of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem Review of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem Steven J. Miller June 19, 2004 Abstract Matrices can be thought of as rectangular (often square) arrays of numbers, or as

More information

MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra

MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra A. Vectors A vector is a quantity that has both magnitude and direction, like velocity. The location of a vector is irrelevant;

More information

4. Determinants.

4. Determinants. 4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.

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

Matrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A =

Matrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A = 30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A = a 11 a 12... a 1N a 21 a 22... a 2N...... a M1 a M2... a MN A matrix can

More information

Mathematical Foundations

Mathematical Foundations Chapter 1 Mathematical Foundations 1.1 Big-O Notations In the description of algorithmic complexity, we often have to use the order notations, often in terms of big O and small o. Loosely speaking, for

More information

Multiple View Geometry in Computer Vision

Multiple View Geometry in Computer Vision Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Scene Planes & Homographies Lecture 19 March 24, 2005 2 In our last lecture, we examined various

More information

Jordan Canonical Form

Jordan Canonical Form Jordan Canonical Form Massoud Malek Jordan normal form or Jordan canonical form (named in honor of Camille Jordan) shows that by changing the basis, a given square matrix M can be transformed into a certain

More information

Singular Value Decomposition

Singular 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 information

1 Linearity and Linear Systems

1 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 information

B553 Lecture 5: Matrix Algebra Review

B553 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 information

Lecture 2: Vector-Vector Operations

Lecture 2: Vector-Vector Operations Lecture 2: Vector-Vector Operations Vector-Vector Operations Addition of two vectors Geometric representation of addition and subtraction of vectors Vectors and points Dot product of two vectors Geometric

More information

L3: Review of linear algebra and MATLAB

L3: Review of linear algebra and MATLAB L3: Review of linear algebra and MATLAB Vector and matrix notation Vectors Matrices Vector spaces Linear transformations Eigenvalues and eigenvectors MATLAB primer CSCE 666 Pattern Analysis Ricardo Gutierrez-Osuna

More information

Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij

Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix

More information

Basic Concepts in Matrix Algebra

Basic Concepts in Matrix Algebra Basic Concepts in Matrix Algebra An column array of p elements is called a vector of dimension p and is written as x p 1 = x 1 x 2. x p. The transpose of the column vector x p 1 is row vector x = [x 1

More information

Principal Component Analysis (PCA)

Principal Component Analysis (PCA) Principal Component Analysis (PCA) Salvador Dalí, Galatea of the Spheres CSC411/2515: Machine Learning and Data Mining, Winter 2018 Michael Guerzhoy and Lisa Zhang Some slides from Derek Hoiem and Alysha

More information

The Cartan Dieudonné Theorem

The Cartan Dieudonné Theorem Chapter 7 The Cartan Dieudonné Theorem 7.1 Orthogonal Reflections Orthogonal symmetries are a very important example of isometries. First let us review the definition of a (linear) projection. Given a

More information

Study Notes on Matrices & Determinants for GATE 2017

Study Notes on Matrices & Determinants for GATE 2017 Study Notes on Matrices & Determinants for GATE 2017 Matrices and Determinates are undoubtedly one of the most scoring and high yielding topics in GATE. At least 3-4 questions are always anticipated from

More information

Solutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015

Solutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015 Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See

More information

Chapter 8. Rigid transformations

Chapter 8. Rigid transformations Chapter 8. Rigid transformations We are about to start drawing figures in 3D. There are no built-in routines for this purpose in PostScript, and we shall have to start more or less from scratch in extending

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

Linear algebra II Homework #1 solutions A = This means that every eigenvector with eigenvalue λ = 1 must have the form

Linear algebra II Homework #1 solutions A = This means that every eigenvector with eigenvalue λ = 1 must have the form Linear algebra II Homework # solutions. Find the eigenvalues and the eigenvectors of the matrix 4 6 A =. 5 Since tra = 9 and deta = = 8, the characteristic polynomial is f(λ) = λ (tra)λ+deta = λ 9λ+8 =

More information

MATH 369 Linear Algebra

MATH 369 Linear Algebra Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine

More information

Final Exam. Linear Algebra Summer 2011 Math S2010X (3) Corrin Clarkson. August 10th, Solutions

Final Exam. Linear Algebra Summer 2011 Math S2010X (3) Corrin Clarkson. August 10th, Solutions Final Exam Linear Algebra Summer Math SX (3) Corrin Clarkson August th, Name: Solutions Instructions: This is a closed book exam. You may not use the textbook, notes or a calculator. You will have 9 minutes

More information

Linear Algebra: Characteristic Value Problem

Linear Algebra: Characteristic Value Problem Linear Algebra: Characteristic Value Problem . The Characteristic Value Problem Let < be the set of real numbers and { be the set of complex numbers. Given an n n real matrix A; does there exist a number

More information

c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

c c c c c c c c c c a 3x3 matrix C= has a determinant determined by Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.

More information

MATH 304 Linear Algebra Lecture 34: Review for Test 2.

MATH 304 Linear Algebra Lecture 34: Review for Test 2. MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1

More information

COMP6237 Data Mining Covariance, EVD, PCA & SVD. Jonathon Hare

COMP6237 Data Mining Covariance, EVD, PCA & SVD. Jonathon Hare COMP6237 Data Mining Covariance, EVD, PCA & SVD Jonathon Hare jsh2@ecs.soton.ac.uk Variance and Covariance Random Variables and Expected Values Mathematicians talk variance (and covariance) in terms of

More information

Examples and MatLab. Vector and Matrix Material. Matrix Addition R = A + B. Matrix Equality A = B. Matrix Multiplication R = A * B.

Examples and MatLab. Vector and Matrix Material. Matrix Addition R = A + B. Matrix Equality A = B. Matrix Multiplication R = A * B. Vector and Matrix Material Examples and MatLab Matrix = Rectangular array of numbers, complex numbers or functions If r rows & c columns, r*c elements, r*c is order of matrix. A is n * m matrix Square

More information

Linear Algebra- Final Exam Review

Linear Algebra- Final Exam Review Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.

More information

Spring, 2012 CIS 515. Fundamentals of Linear Algebra and Optimization Jean Gallier

Spring, 2012 CIS 515. Fundamentals of Linear Algebra and Optimization Jean Gallier Spring 0 CIS 55 Fundamentals of Linear Algebra and Optimization Jean Gallier Homework 5 & 6 + Project 3 & 4 Note: Problems B and B6 are for extra credit April 7 0; Due May 7 0 Problem B (0 pts) Let A be

More information

LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY

LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT 204 - FALL 2006 PRINCETON UNIVERSITY ALFONSO SORRENTINO 1 Adjoint of a linear operator Note: In these notes, V will denote a n-dimensional euclidean vector

More information

Lecture 3: Review of Linear Algebra

Lecture 3: Review of Linear Algebra ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters, transforms,

More information

Dot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.

Dot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Dot Products K. Behrend April 3, 008 Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Contents The dot product 3. Length of a vector........................

More information

Lecture 3: Review of Linear Algebra

Lecture 3: Review of Linear Algebra ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak, scribe: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters,

More information

MLCC 2015 Dimensionality Reduction and PCA

MLCC 2015 Dimensionality Reduction and PCA MLCC 2015 Dimensionality Reduction and PCA Lorenzo Rosasco UNIGE-MIT-IIT June 25, 2015 Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component

More information

CS168: The Modern Algorithmic Toolbox Lecture #8: How PCA Works

CS168: The Modern Algorithmic Toolbox Lecture #8: How PCA Works CS68: The Modern Algorithmic Toolbox Lecture #8: How PCA Works Tim Roughgarden & Gregory Valiant April 20, 206 Introduction Last lecture introduced the idea of principal components analysis (PCA). The

More information

1 Planar rotations. Math Abstract Linear Algebra Fall 2011, section E1 Orthogonal matrices and rotations

1 Planar rotations. Math Abstract Linear Algebra Fall 2011, section E1 Orthogonal matrices and rotations Math 46 - Abstract Linear Algebra Fall, section E Orthogonal matrices and rotations Planar rotations Definition: A planar rotation in R n is a linear map R: R n R n such that there is a plane P R n (through

More information

Exercise Sheet 1. 1 Probability revision 1: Student-t as an infinite mixture of Gaussians

Exercise Sheet 1. 1 Probability revision 1: Student-t as an infinite mixture of Gaussians Exercise Sheet 1 1 Probability revision 1: Student-t as an infinite mixture of Gaussians Show that an infinite mixture of Gaussian distributions, with Gamma distributions as mixing weights in the following

More information

Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )

Ir 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 information

Intro Vectors 2D implicit curves 2D parametric curves. Graphics 2011/2012, 4th quarter. Lecture 2: vectors, curves, and surfaces

Intro Vectors 2D implicit curves 2D parametric curves. Graphics 2011/2012, 4th quarter. Lecture 2: vectors, curves, and surfaces Lecture 2, curves, and surfaces Organizational remarks Tutorials: Tutorial 1 will be online later today TA sessions for questions start next week Practicals: Exams: Make sure to find a team partner very

More information

Definition (T -invariant subspace) Example. Example

Definition (T -invariant subspace) Example. Example Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin

More information

Math Linear Algebra II. 1. Inner Products and Norms

Math Linear Algebra II. 1. Inner Products and Norms Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,

More information

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix

More information

Section 13.4 The Cross Product

Section 13.4 The Cross Product Section 13.4 The Cross Product Multiplying Vectors 2 In this section we consider the more technical multiplication which can be defined on vectors in 3-space (but not vectors in 2-space). 1. Basic Definitions

More information

Topic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C

Topic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C Topic 1 Quiz 1 text A reduced row-echelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correct-choice may have 0, 1, 2, or 3 choice may have 0,

More information

Principle Components Analysis (PCA) Relationship Between a Linear Combination of Variables and Axes Rotation for PCA

Principle Components Analysis (PCA) Relationship Between a Linear Combination of Variables and Axes Rotation for PCA Principle Components Analysis (PCA) Relationship Between a Linear Combination of Variables and Axes Rotation for PCA Principle Components Analysis: Uses one group of variables (we will call this X) In

More information

Online Exercises for Linear Algebra XM511

Online Exercises for Linear Algebra XM511 This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2

More information