Eigenvalues, Eigenvectors, and an Intro to PCA
|
|
- Joel O’Brien’
- 6 years ago
- Views:
Transcription
1 Eigenvalues, Eigenvectors, and an Intro to PCA Eigenvalues, Eigenvectors, and an Intro to PCA
2 Changing Basis We ve talked so far about re-writing our data using a new set of variables, or a new basis. How to choose this new basis?? Eigenvalues, Eigenvectors, and an Intro to PCA
3 PCA One (very popular) method: start by choosing the basis vectors as the directions in which the variance of the data is maximal. weight height Eigenvalues, Eigenvectors, and an Intro to PCA
4 PCA One (very popular) method: start by choosing the basis vectors as the directions in which the variance of the data is maximal. v1 w h Eigenvalues, Eigenvectors, and an Intro to PCA
5 PCA Then, choose subsequent directions that are orthogonal to the first and have the next largest variance. v2 w h Eigenvalues, Eigenvectors, and an Intro to PCA
6 PCA As it turns out, these directions are given by the eigenvectors of either: The Covariance Matrix (data is only centered) The Correlation Matrix (data is completely standardized) Eigenvalues, Eigenvectors, and an Intro to PCA
7 PCA As it turns out, these directions are given by the eigenvectors of either: The Covariance Matrix (data is only centered) The Correlation Matrix (data is completely standardized) Great. What the heck is an eigenvector? Eigenvalues, Eigenvectors, and an Intro to PCA
8 Eigenvalues and Eigenvectors
9 Effect of Matrix Multiplication When we multiply a matrix by a vector, what do we get? A x =? =
10 Effect of Matrix Multiplication When we multiply a matrix by a vector, what do we get? A x =? = b Another vector.
11 Effect of Matrix Multiplication When we multiply a matrix by a vector, what do we get? A x =? = b Another vector. When the matrix A is square (n n) then x and b are both vectors in R n. We can draw (or imagine) them in the same space.
12 Linear Transformation In general, multiplying a vector by a matrix changes both its magnitude and direction. ( ) ( ) A = x = Ax = 6 0 5
13 Linear Transformation In general, multiplying a vector by a matrix changes both its magnitude and direction. ( ) ( ) ( ) A = x = Ax = span(e2) x Ax span(e1)
14 Eigenvectors However, a matrix may act on certain vectors by changing only their magnitude, not their direction (or possibly reversing it) ( ) ( ) A = x = Ax = 6 0 3
15 Eigenvectors However, a matrix may act on certain vectors by changing only their magnitude, not their direction (or possibly reversing it) ( ) ( ) ( ) A = x = Ax = span(e2) Ax x span(e1) The matrix A acts like a scalar!
16 Eigenvalues and Eigenvectors For a square matrix, A, a non-zero vector x is called an eigenvector of A if multiplying x by A results in a scalar multiple of x. Ax = λx
17 Eigenvalues and Eigenvectors For a square matrix, A, a non-zero vector x is called an eigenvector of A if multiplying x by A results in a scalar multiple of x. Ax = λx The scalar λ is called the eigenvalue associated with the eigenvector.
18 Previous Example ( ) 1 1 A = 6 0 ( ) 2 Ax = = 6 x = ( ) 1 3
19 Previous Example ( ) 1 1 A = 6 0 ( ) 2 Ax = = 2 6 x = ( ) 1 3 ( ) 1 = 2x 3 = λ = 2 is the eigenvalue span(e2) Ax x span(e1)
20 Example 2 Show that x is an eigenvector of A and find the corresponding eigenvalue. ( ) ( ) A = x = 6 0 2
21 Example 2 Show that x is an eigenvector of A and find the corresponding eigenvalue. ( ) ( ) A = x = ( ) 3 Ax = ( 1 = ) = 3x
22 Example 2 Show that x is an eigenvector of A and find the corresponding eigenvalue. ( ) ( ) A = x = ( ) 3 Ax = ( 1 = ) = 3x = λ = 3 is the eigenvalue corresponding to the eigenvector x
23 Example 2 A = ( ) x = span(e2) ( ) 1 2 Ax = 3x Ax x span(e1)
24 Let s Practice 1 Show that v is an eigenvector of A and find the corresponding eigenvalue: A = ( ) v = ( ) 3 3
25 Let s Practice 1 Show that v is an eigenvector of A and find the corresponding eigenvalue: A = ( ) v = ( ) 3 3 A = ( ) v = ( ) 1 2
26 Let s Practice 1 Show that v is an eigenvector of A and find the corresponding eigenvalue: A = ( ) v = ( ) 3 3 A = ( ) v = ( ) Can a rectangular matrix have eigenvalues/eigenvectors?
27 Eigenspaces Fact: Any scalar multiple of an eigenvector of A is also an eigenvector of A with the same eigenvalue: A = ( ) x = ( ) 1 3 λ = 2 Try v = ( ) 2 6 or u = ( ) 1 3 or z = ( ) 3 9
28 Eigenspaces Fact: Any scalar multiple of an eigenvector of A is also an eigenvector of A with the same eigenvalue: A = ( ) x = ( ) 1 3 λ = 2 Try v = ( ) 2 6 or u = ( ) 1 3 or z = ( ) 3 9 In general (#proof): Let Ax = λx. If c is some constant, then: A(cx) = c(ax) = c(λx) = λ(cx)
29 Eigenspaces For a given matrix, there are infinitely many eigenvectors associated with any one eigenvalue.
30 Eigenspaces For a given matrix, there are infinitely many eigenvectors associated with any one eigenvalue. Any scalar multiple (positive or negative) can be used.
31 Eigenspaces For a given matrix, there are infinitely many eigenvectors associated with any one eigenvalue. Any scalar multiple (positive or negative) can be used. The collection is referred to as the eigenspace associated with the eigenvalue.
32 Eigenspaces For a given matrix, there are infinitely many eigenvectors associated with any one eigenvalue. Any scalar multiple (positive or negative) can be used. The collection is referred to as the eigenspace associated with the eigenvalue. In the previous example, the eigenspace of λ = 2 was the { span x = ( )} 1. 3
33 Eigenspaces For a given matrix, there are infinitely many eigenvectors associated with any one eigenvalue. Any scalar multiple (positive or negative) can be used. The collection is referred to as the eigenspace associated with the eigenvalue. In the previous example, the eigenspace of λ = 2 was the { span x = ( )} 1. 3 With this in mind what should you expect from software?
34 Zero Eigenvalues What if λ = 0 is an eigenvalue for some matrix A?
35 Zero Eigenvalues What if λ = 0 is an eigenvalue for some matrix A? Ax = 0x = 0, where x 0 is an eigenvector.
36 Zero Eigenvalues What if λ = 0 is an eigenvalue for some matrix A? Ax = 0x = 0, where x 0 is an eigenvector. This means that some linear combination of the columns of A equals zero! = Columns of A are linearly dependent.
37 Zero Eigenvalues What if λ = 0 is an eigenvalue for some matrix A? Ax = 0x = 0, where x 0 is an eigenvector. This means that some linear combination of the columns of A equals zero! = Columns of A are linearly dependent. = A is not full rank.
38 Zero Eigenvalues What if λ = 0 is an eigenvalue for some matrix A? Ax = 0x = 0, where x 0 is an eigenvector. This means that some linear combination of the columns of A equals zero! = Columns of A are linearly dependent. = A is not full rank. = Perfect Multicollinearity.
39 Zero Eigenvalues What if λ = 0 is an eigenvalue for some matrix A? Ax = 0x = 0, where x 0 is an eigenvector. This means that some linear combination of the columns of A equals zero! = Columns of A are linearly dependent. = A is not full rank. = Perfect Multicollinearity.
40 Eigenpairs For a square (n n) matrix A, eigenvalues and eigenvectors come in pairs. Normally, the pairs are ordered by magnitude of eigenvalue. λ 1 λ 2 λ 3 λ n λ 1 is the largest eigenvalue Eigenvector v i corresponds to eigenvalue λ i
41 Let s Practice 1 For the following matrix, determine the eigenvalue associated with the given eigenvector A = v = From this eigenvalue, what can you conclude about the matrix A?
42 Let s Practice 1 For the following matrix, determine the eigenvalue associated with the given eigenvector A = v = From this eigenvalue, what can you conclude about the matrix A? 2 The matrix M has eigenvectors u and v. What is λ 1, the first eigenvalue for the matrix M? M = ( ) u = ( 1 3 ) v = ( ) 1 2
43 Let s Practice 1 For the following matrix, determine the eigenvalue associated with the given eigenvector A = v = From this eigenvalue, what can you conclude about the matrix A? 2 The matrix M has eigenvectors u and v. What is λ 1, the first eigenvalue for the matrix M? M = ( ) u = ( 1 3 ) v = ( ) For the previous problem, is the specific eigenvector (u or v) the only eigenvector associated with λ 1?
44 Diagonalization What happens if I put all the eigenvectors of A into a matrix as columns: V = [v 1 v 2 v 3... v n ] and then compute AV?
45 Diagonalization What happens if I put all the eigenvectors of A into a matrix as columns: V = [v 1 v 2 v 3... v n ] and then compute AV? AV = A[v 1 v 2 v 3... v n ]
46 Diagonalization What happens if I put all the eigenvectors of A into a matrix as columns: V = [v 1 v 2 v 3... v n ] and then compute AV? AV = A[v 1 v 2 v 3... v n ] = [Av 1 Av 2 Av 3... Av n ]
47 Diagonalization What happens if I put all the eigenvectors of A into a matrix as columns: V = [v 1 v 2 v 3... v n ] and then compute AV? AV = A[v 1 v 2 v 3... v n ] = [Av 1 Av 2 Av 3... Av n ] = [λ 1 v 1 λ 2 v 2 λ 3 v 3... λ n v n ]
48 Diagonalization What happens if I put all the eigenvectors of A into a matrix as columns: V = [v 1 v 2 v 3... v n ] and then compute AV? AV = A[v 1 v 2 v 3... v n ] = [Av 1 Av 2 Av 3... Av n ] = [λ 1 v 1 λ 2 v 2 λ 3 v 3... λ n v n ] The effect is that the columns are all scaled by the corresponding eigenvalue.
49 Diagonalization AV = [λ 1 v 1 λ 2 v 2 λ 3 v 3... λ n v n ] The same thing would happen if we multiplied V by a diagonal matrix of eigenvalues on the right: λ λ VD = [v 1 v 2 v 3... v n ] λ n = [λ 1 v 1 λ 2 v 2 λ 3 v 3... λ n v n ]
50 Diagonalization AV = [λ 1 v 1 λ 2 v 2 λ 3 v 3... λ n v n ] The same thing would happen if we multiplied V by a diagonal matrix of eigenvalues on the right: λ λ VD = [v 1 v 2 v 3... v n ] λ n = [λ 1 v 1 λ 2 v 2 λ 3 v 3... λ n v n ] = AV = VD
51 Diagonalization AV = VD
52 Diagonalization AV = VD If the matrix A has n linearly independent eigenvectors, then the matrix V has an inverse.
53 Diagonalization AV = VD If the matrix A has n linearly independent eigenvectors, then the matrix V has an inverse. = V 1 AV = D
54 Diagonalization AV = VD If the matrix A has n linearly independent eigenvectors, then the matrix V has an inverse. = V 1 AV = D This is called the diagonalization of A.
55 Diagonalization AV = VD If the matrix A has n linearly independent eigenvectors, then the matrix V has an inverse. = V 1 AV = D This is called the diagonalization of A. It is only possible when A has n linearly independent eigenvectors (so that the matrix V has an inverse).
56 Let s Practice For the matrix A = ( ) 0 4, 1 5 a. The pairs λ 1 = 4, v 1 = ( ) 1 1 and λ 2 = 1, v 2 = ( ) 4 1 are eigenpairs of A. For each pair, provide another eigenvector associated with that eigenvalue.
57 Let s Practice For the matrix A = ( ) 0 4, 1 5 a. The pairs λ 1 = 4, v 1 = ( ) 1 1 and λ 2 = 1, v 2 = ( ) 4 1 are eigenpairs of A. For each pair, provide another eigenvector associated with that eigenvalue. b. Based on the eigenvectors listed in part a, can the matrix A be diagonalized? Why or why not? If diagonalization is possible, explain how it would be done.
58 Symmetric Matrices - Properties Symmetric matrices (like the covariance matrix, the correlation matrix, and X T X) have some very nice properties:
59 Symmetric Matrices - Properties Symmetric matrices (like the covariance matrix, the correlation matrix, and X T X) have some very nice properties: 1 They are always diagonalizable. (Always have n linearly independent eigenvectors.)
60 Symmetric Matrices - Properties Symmetric matrices (like the covariance matrix, the correlation matrix, and X T X) have some very nice properties: 1 They are always diagonalizable. (Always have n linearly independent eigenvectors.) 2 Their eigenvectors are all mutually orthogonal.
61 Symmetric Matrices - Properties Symmetric matrices (like the covariance matrix, the correlation matrix, and X T X) have some very nice properties: 1 They are always diagonalizable. (Always have n linearly independent eigenvectors.) 2 Their eigenvectors are all mutually orthogonal. 3 Thus, if you normalize the eigenvectors (software does this automatically) they will form an orthogonal matrix, V.
62 Symmetric Matrices - Properties Symmetric matrices (like the covariance matrix, the correlation matrix, and X T X) have some very nice properties: 1 They are always diagonalizable. (Always have n linearly independent eigenvectors.) 2 Their eigenvectors are all mutually orthogonal. 3 Thus, if you normalize the eigenvectors (software does this automatically) they will form an orthogonal matrix, V. = AV = VD
63 Symmetric Matrices - Properties Symmetric matrices (like the covariance matrix, the correlation matrix, and X T X) have some very nice properties: 1 They are always diagonalizable. (Always have n linearly independent eigenvectors.) 2 Their eigenvectors are all mutually orthogonal. 3 Thus, if you normalize the eigenvectors (software does this automatically) they will form an orthogonal matrix, V. = AV = VD V T AV = D
64 Eigenvectors of Symmetric Matrices
65 Introduction to Principal Components Analysis (PCA) Eigenvalues and Eigenvectors Introduction to Principal Components Analysis (PCA)
66 Eigenvectors of the Covariance/Correlation Matrices Introduction to Principal Components Analysis (PCA)
67 Eigenvectors of the Covariance/Correlation Matrices Covariance/Correlation matrices are symmetric. Introduction to Principal Components Analysis (PCA)
68 Eigenvectors of the Covariance/Correlation Matrices Covariance/Correlation matrices are symmetric. = Eigenvectors are orthogonal Introduction to Principal Components Analysis (PCA)
69 Eigenvectors of the Covariance/Correlation Matrices Covariance/Correlation matrices are symmetric. = Eigenvectors are orthogonal Eigenvectors are ordered by the magnitude of eigenvalues: λ 1 λ 2 λ p {v 1, v 2,..., v n } Introduction to Principal Components Analysis (PCA)
70 Covariance vs. Correlation We ll get into more detail later, but Introduction to Principal Components Analysis (PCA)
71 Covariance vs. Correlation We ll get into more detail later, but For the covariance matrix, we want to think of our data as centered to begin with. Introduction to Principal Components Analysis (PCA)
72 Covariance vs. Correlation We ll get into more detail later, but For the covariance matrix, we want to think of our data as centered to begin with. For correlation matrix, we want to think of our data as standardized to begin with. (i.e. centered and divided by standard deviation). Introduction to Principal Components Analysis (PCA)
73 Direction of Maximal Variance The first eigenvector of a covariance/correlation matrix points in the direction of maximum variance in the data. This eigenvector is the first principal component. weight height Introduction to Principal Components Analysis (PCA)
74 Direction of Maximal Variance The first eigenvector of a covariance matrix points in the direction of maximum variance in the data. This eigenvector is the first principal component. v1 w h Introduction to Principal Components Analysis (PCA)
75 Not a Regression Line The first principal component is not the same direction as the regression line. It minimizes orthogonal distances, not vertical distances. v1 w h Introduction to Principal Components Analysis (PCA)
76 Not a Regression Line It may be close in some cases, but it s not the same. x2 Principal Component Regression Line x1 Introduction to Principal Components Analysis (PCA)
77 Secondary Directions The second eigenvector of a covariance matrix points in the direction, orthogonal to the first, that has the maximum variance. v2 w h Introduction to Principal Components Analysis (PCA)
78 A New Basis Principal components provide us with a new orthogonal basis where the new coordinates of the data points are uncorrelated: Introduction to Principal Components Analysis (PCA)
79 Variable Loadings Each principal component is a linear combination of the original variables. ( ) 0.7 v 1 = = 0.7h + 0.7w 0.7 ( ) 0.7 v 2 = = 0.7h + 0.7w 0.7 These coefficients are called loadings (factor loadings). v2 w h Introduction to Principal Components Analysis (PCA)
80 Combining PCs Likewise, we can think of our original variables as linear combinations of the principal components with coordinates in the new basis: ( ) 0.7 h = = 0.7v v 2 ( ) 0.7 w = = 0.7v v 2 w h Introduction to Principal Components Analysis (PCA)
81 The Biplot Uncorrelated data points and variable vectors plotted on the same set of axis! Points in top right have largest weight values Points in top left have smallest height values w h Introduction to Principal Components Analysis (PCA)
82 weight height Introduction to Principal Components Analysis (PCA)
83 Variable Loadings The variable loadings give us a formula for finding the coordinates of our data in the new basis. ( ) 0.7 v 1 = = PC = 0.7height + 0.7weight v 2 = ( ) 0.7 = PC = 0.7height + 0.7weight From these loadings, we can get a sense of how important each original variable is to our new variables (the components). Introduction to Principal Components Analysis (PCA)
84 Let s Practice 1 For the following data plot, take your best guess and draw the directions of the first and second principal components. 2 Is there more than one answer to this question? Introduction to Principal Components Analysis (PCA)
85 Let s Practice Suppose your data contained the variables VO2_max, mile pace, and weight in that order. The first principal component for this data is the eigenvector of the covariance matrix What variable is most important/influential to the first principal component? Introduction to Principal Components Analysis (PCA)
Eigenvalues, Eigenvectors, and an Intro to PCA
Eigenvalues, Eigenvectors, and an Intro to PCA Eigenvalues, Eigenvectors, and an Intro to PCA Changing Basis We ve talked so far about re-writing our data using a new set of variables, or a new basis.
More informationEigenvalues, Eigenvectors, and an Intro to PCA
Eigenvalues, Eigenvectors, and an Intro to PCA Eigenvalues, Eigenvectors, and an Intro to PCA Changing Basis We ve talked so far about re-writing our data using a new set of variables, or a new basis.
More informationDIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix
DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that
More informationLecture 15, 16: Diagonalization
Lecture 15, 16: Diagonalization Motivation: Eigenvalues and Eigenvectors are easy to compute for diagonal matrices. Hence, we would like (if possible) to convert matrix A into a diagonal matrix. Suppose
More informationPRINCIPAL COMPONENTS ANALYSIS
121 CHAPTER 11 PRINCIPAL COMPONENTS ANALYSIS We now have the tools necessary to discuss one of the most important concepts in mathematical statistics: Principal Components Analysis (PCA). PCA involves
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationSingular Value Decomposition. 1 Singular Value Decomposition and the Four Fundamental Subspaces
Singular Value Decomposition This handout is a review of some basic concepts in linear algebra For a detailed introduction, consult a linear algebra text Linear lgebra and its pplications by Gilbert Strang
More informationICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization
ICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 21 Symmetric matrices An n n
More informationSolutions to Final Exam
Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5-vector. Assume that the nullity of A is. (a) What is the rank of A? 3 (b) Are the rows of A linearly independent? (c) Are the columns
More informationVAR Model. (k-variate) VAR(p) model (in the Reduced Form): Y t-2. Y t-1 = A + B 1. Y t + B 2. Y t-p. + ε t. + + B p. where:
VAR Model (k-variate VAR(p model (in the Reduced Form: where: Y t = A + B 1 Y t-1 + B 2 Y t-2 + + B p Y t-p + ε t Y t = (y 1t, y 2t,, y kt : a (k x 1 vector of time series variables A: a (k x 1 vector
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationDiagonalization of Matrix
of Matrix King Saud University August 29, 2018 of Matrix Table of contents 1 2 of Matrix Definition If A M n (R) and λ R. We say that λ is an eigenvalue of the matrix A if there is X R n \ {0} such that
More informationMATH 221, Spring Homework 10 Solutions
MATH 22, Spring 28 - Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the
More informationComputationally, diagonal matrices are the easiest to work with. With this idea in mind, we introduce similarity:
Diagonalization We have seen that diagonal and triangular matrices are much easier to work with than are most matrices For example, determinants and eigenvalues are easy to compute, and multiplication
More informationKnowledge Discovery and Data Mining 1 (VO) ( )
Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory
More informationPrinciple 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 informationEigenvalues, Eigenvectors, and Diagonalization
Math 240 TA: Shuyi Weng Winter 207 February 23, 207 Eigenvalues, Eigenvectors, and Diagonalization The concepts of eigenvalues, eigenvectors, and diagonalization are best studied with examples. We will
More informationThe Jordan Normal Form and its Applications
The and its Applications Jeremy IMPACT Brigham Young University A square matrix A is a linear operator on {R, C} n. A is diagonalizable if and only if it has n linearly independent eigenvectors. What happens
More informationMATH 583A REVIEW SESSION #1
MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),
More informationPrincipal Components Analysis (PCA)
Principal Components Analysis (PCA) Principal Components Analysis (PCA) a technique for finding patterns in data of high dimension Outline:. Eigenvectors and eigenvalues. PCA: a) Getting the data b) Centering
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 informationMAT 1302B Mathematical Methods II
MAT 1302B Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2015 Lecture 19 Alistair Savage (uottawa) MAT 1302B Mathematical Methods II Winter 2015 Lecture
More informationMathematical foundations - linear algebra
Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar
More informationEcon Slides from Lecture 7
Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for
More informationLecture 13. Principal Component Analysis. Brett Bernstein. April 25, CDS at NYU. Brett Bernstein (CDS at NYU) Lecture 13 April 25, / 26
Principal Component Analysis Brett Bernstein CDS at NYU April 25, 2017 Brett Bernstein (CDS at NYU) Lecture 13 April 25, 2017 1 / 26 Initial Question Intro Question Question Let S R n n be symmetric. 1
More informationHomework For each of the following matrices, find the minimal polynomial and determine whether the matrix is diagonalizable.
Math 5327 Fall 2018 Homework 7 1. For each of the following matrices, find the minimal polynomial and determine whether the matrix is diagonalizable. 3 1 0 (a) A = 1 2 0 1 1 0 x 3 1 0 Solution: 1 x 2 0
More informationHomework sheet 4: EIGENVALUES AND EIGENVECTORS. DIAGONALIZATION (with solutions) Year ? Why or why not? 6 9
Bachelor in Statistics and Business Universidad Carlos III de Madrid Mathematical Methods II María Barbero Liñán Homework sheet 4: EIGENVALUES AND EIGENVECTORS DIAGONALIZATION (with solutions) Year - Is
More informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More informationPrincipal Component Analysis (PCA) Principal Component Analysis (PCA)
Recall: Eigenvectors of the Covariance Matrix Covariance matrices are symmetric. Eigenvectors are orthogonal Eigenvectors are ordered by the magnitude of eigenvalues: λ 1 λ 2 λ p {v 1, v 2,..., v n } Recall:
More information1 Principal Components Analysis
Lecture 3 and 4 Sept. 18 and Sept.20-2006 Data Visualization STAT 442 / 890, CM 462 Lecture: Ali Ghodsi 1 Principal Components Analysis Principal components analysis (PCA) is a very popular technique for
More informationPrincipal Component Analysis. Applied Multivariate Statistics Spring 2012
Principal Component Analysis Applied Multivariate Statistics Spring 2012 Overview Intuition Four definitions Practical examples Mathematical example Case study 2 PCA: Goals Goal 1: Dimension reduction
More informationIntelligent Data Analysis. Principal Component Analysis. School of Computer Science University of Birmingham
Intelligent Data Analysis Principal Component Analysis Peter Tiňo School of Computer Science University of Birmingham Discovering low-dimensional spatial layout in higher dimensional spaces - 1-D/3-D example
More informationJordan Canonical Form Homework Solutions
Jordan Canonical Form Homework Solutions For each of the following, put the matrix in Jordan canonical form and find the matrix S such that S AS = J. [ ]. A = A λi = λ λ = ( λ) = λ λ = λ =, Since we have
More informationDiagonalization of Matrices
LECTURE 4 Diagonalization of Matrices Recall that a diagonal matrix is a square n n matrix with non-zero entries only along the diagonal from the upper left to the lower right (the main diagonal) Diagonal
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationMAC Module 12 Eigenvalues and Eigenvectors. Learning Objectives. Upon completing this module, you should be able to:
MAC Module Eigenvalues and Eigenvectors Learning Objectives Upon completing this module, you should be able to: Solve the eigenvalue problem by finding the eigenvalues and the corresponding eigenvectors
More informationMAC Module 12 Eigenvalues and Eigenvectors
MAC 23 Module 2 Eigenvalues and Eigenvectors Learning Objectives Upon completing this module, you should be able to:. Solve the eigenvalue problem by finding the eigenvalues and the corresponding eigenvectors
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 information8 Eigenvectors and the Anisotropic Multivariate Gaussian Distribution
Eigenvectors and the Anisotropic Multivariate Gaussian Distribution Eigenvectors and the Anisotropic Multivariate Gaussian Distribution EIGENVECTORS [I don t know if you were properly taught about eigenvectors
More informationPrincipal Component Analysis
Principal Component Analysis Yuanzhen Shao MA 26500 Yuanzhen Shao PCA 1 / 13 Data as points in R n Assume that we have a collection of data in R n. x 11 x 21 x 12 S = {X 1 =., X x 22 2 =.,, X x m2 m =.
More informationOrthogonality. Orthonormal Bases, Orthogonal Matrices. Orthogonality
Orthonormal Bases, Orthogonal Matrices The Major Ideas from Last Lecture Vector Span Subspace Basis Vectors Coordinates in different bases Matrix Factorization (Basics) The Major Ideas from Last Lecture
More informationChapter 4 & 5: Vector Spaces & Linear Transformations
Chapter 4 & 5: Vector Spaces & Linear Transformations Philip Gressman University of Pennsylvania Philip Gressman Math 240 002 2014C: Chapters 4 & 5 1 / 40 Objective The purpose of Chapter 4 is to think
More informationLinear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4
Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am - :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary
More informationMatrices related to linear transformations
Math 4326 Fall 207 Matrices related to linear transformations We have encountered several ways in which matrices relate to linear transformations. In this note, I summarize the important facts and formulas
More informationFinal Exam Practice Problems Answers Math 24 Winter 2012
Final Exam Practice Problems Answers Math 4 Winter 0 () The Jordan product of two n n matrices is defined as A B = (AB + BA), where the products inside the parentheses are standard matrix product. Is the
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationMath 2331 Linear Algebra
5. Eigenvectors & Eigenvalues Math 233 Linear Algebra 5. Eigenvectors & Eigenvalues Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ Shang-Huan Chiu,
More informationIntroduction to Machine Learning
10-701 Introduction to Machine Learning PCA Slides based on 18-661 Fall 2018 PCA Raw data can be Complex, High-dimensional To understand a phenomenon we measure various related quantities If we knew what
More information1. In this problem, if the statement is always true, circle T; otherwise, circle F.
Math 1553, Extra Practice for Midterm 3 (sections 45-65) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation
More informationMore Linear Algebra. Edps/Soc 584, Psych 594. Carolyn J. Anderson
More Linear Algebra Edps/Soc 584, Psych 594 Carolyn J. Anderson Department of Educational Psychology I L L I N O I S university of illinois at urbana-champaign c Board of Trustees, University of Illinois
More informationVectors and Matrices Statistics with Vectors and Matrices
Vectors and Matrices Statistics with Vectors and Matrices Lecture 3 September 7, 005 Analysis Lecture #3-9/7/005 Slide 1 of 55 Today s Lecture Vectors and Matrices (Supplement A - augmented with SAS proc
More informationLINEAR 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 information7. Symmetric Matrices and Quadratic Forms
Linear Algebra 7. Symmetric Matrices and Quadratic Forms CSIE NCU 1 7. Symmetric Matrices and Quadratic Forms 7.1 Diagonalization of symmetric matrices 2 7.2 Quadratic forms.. 9 7.4 The singular value
More informationFirst of all, the notion of linearity does not depend on which coordinates are used. Recall that a map T : R n R m is linear if
5 Matrices in Different Coordinates In this section we discuss finding matrices of linear maps in different coordinates Earlier in the class was the matrix that multiplied by x to give ( x) in standard
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationAppendix A: Matrices
Appendix A: Matrices A matrix is a rectangular array of numbers Such arrays have rows and columns The numbers of rows and columns are referred to as the dimensions of a matrix A matrix with, say, 5 rows
More informationName Solutions Linear Algebra; Test 3. Throughout the test simplify all answers except where stated otherwise.
Name Solutions Linear Algebra; Test 3 Throughout the test simplify all answers except where stated otherwise. 1) Find the following: (10 points) ( ) Or note that so the rows are linearly independent, so
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall 2015 1 / 14 Introduction We define eigenvalues and eigenvectors. We discuss how to
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture 9: Diagonalization Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./9 Section. Diagonalization The goal here is to develop a useful
More informationMATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization.
MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization. Eigenvalues and eigenvectors of an operator Definition. Let V be a vector space and L : V V be a linear operator. A number λ
More informationWarm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions
Warm-up True or false? 1. proj u proj v u = u 2. The system of normal equations for A x = y has solutions iff A x = y has solutions 3. The normal equations are always consistent Baby proof 1. Let A be
More information0 2 0, it is diagonal, hence diagonalizable)
MATH 54 TRUE/FALSE QUESTIONS FOR MIDTERM 2 SOLUTIONS PEYAM RYAN TABRIZIAN 1. (a) TRUE If A is diagonalizable, then A 3 is diagonalizable. (A = P DP 1, so A 3 = P D 3 P = P D P 1, where P = P and D = D
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 informationPrincipal Component Analysis-I Geog 210C Introduction to Spatial Data Analysis. Chris Funk. Lecture 17
Principal Component Analysis-I Geog 210C Introduction to Spatial Data Analysis Chris Funk Lecture 17 Outline Filters and Rotations Generating co-varying random fields Translating co-varying fields into
More informationData Mining Lecture 4: Covariance, EVD, PCA & SVD
Data Mining Lecture 4: Covariance, EVD, PCA & SVD Jo Houghton ECS Southampton February 25, 2019 1 / 28 Variance and Covariance - Expectation A random variable takes on different values due to chance The
More information. = V c = V [x]v (5.1) c 1. c k
Chapter 5 Linear Algebra It can be argued that all of linear algebra can be understood using the four fundamental subspaces associated with a matrix Because they form the foundation on which we later work,
More informationAnnouncements Monday, November 20
Announcements Monday, November 20 You already have your midterms! Course grades will be curved at the end of the semester. The percentage of A s, B s, and C s to be awarded depends on many factors, and
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationAnnouncements Monday, October 29
Announcements Monday, October 29 WeBWorK on determinents due on Wednesday at :59pm. The quiz on Friday covers 5., 5.2, 5.3. My office is Skiles 244 and Rabinoffice hours are: Mondays, 2 pm; Wednesdays,
More informationTMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013
TMA4115 - Calculus 3 Lecture 21, April 3 Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 www.ntnu.no TMA4115 - Calculus 3, Lecture 21 Review of last week s lecture Last week
More informationConcentration Ellipsoids
Concentration Ellipsoids ECE275A Lecture Supplement Fall 2008 Kenneth Kreutz Delgado Electrical and Computer Engineering Jacobs School of Engineering University of California, San Diego VERSION LSECE275CE
More informationCS168: The Modern Algorithmic Toolbox Lecture #7: Understanding Principal Component Analysis (PCA)
CS68: The Modern Algorithmic Toolbox Lecture #7: Understanding Principal Component Analysis (PCA) Tim Roughgarden & Gregory Valiant April 0, 05 Introduction. Lecture Goal Principal components analysis
More informationVectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1. x 2. x =
Linear Algebra Review Vectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1 x x = 2. x n Vectors of up to three dimensions are easy to diagram.
More informationExercises * on Principal Component Analysis
Exercises * on Principal Component Analysis Laurenz Wiskott Institut für Neuroinformatik Ruhr-Universität Bochum, Germany, EU 4 February 207 Contents Intuition 3. Problem statement..........................................
More informationAgenda: Understand the action of A by seeing how it acts on eigenvectors.
Eigenvalues and Eigenvectors If Av=λv with v nonzero, then λ is called an eigenvalue of A and v is called an eigenvector of A corresponding to eigenvalue λ. Agenda: Understand the action of A by seeing
More informationMath 2114 Common Final Exam May 13, 2015 Form A
Math 4 Common Final Exam May 3, 5 Form A Instructions: Using a # pencil only, write your name and your instructor s name in the blanks provided. Write your student ID number and your CRN in the blanks
More informationQuestion 7. Consider a linear system A x = b with 4 unknown. x = [x 1, x 2, x 3, x 4 ] T. The augmented
Question. How many solutions does x 6 = 4 + i have Practice Problems 6 d) 5 Question. Which of the following is a cubed root of the complex number i. 6 e i arctan() e i(arctan() π) e i(arctan() π)/3 6
More informationLinear Algebra. Rekha Santhanam. April 3, Johns Hopkins Univ. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, / 7
Linear Algebra Rekha Santhanam Johns Hopkins Univ. April 3, 2009 Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 1 / 7 Dynamical Systems Denote owl and wood rat populations at time k
More informationMAT1302F Mathematical Methods II Lecture 19
MAT302F Mathematical Methods II Lecture 9 Aaron Christie 2 April 205 Eigenvectors, Eigenvalues, and Diagonalization Now that the basic theory of eigenvalues and eigenvectors is in place most importantly
More informationCOMP6237 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 informationMatrix Vector Products
We covered these notes in the tutorial sessions I strongly recommend that you further read the presented materials in classical books on linear algebra Please make sure that you understand the proofs and
More informationRecitation 9: Probability Matrices and Real Symmetric Matrices. 3 Probability Matrices: Definitions and Examples
Math b TA: Padraic Bartlett Recitation 9: Probability Matrices and Real Symmetric Matrices Week 9 Caltech 20 Random Question Show that + + + + +... = ϕ, the golden ratio, which is = + 5. 2 2 Homework comments
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors week -2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n
More informationCS4495/6495 Introduction to Computer Vision. 8B-L2 Principle Component Analysis (and its use in Computer Vision)
CS4495/6495 Introduction to Computer Vision 8B-L2 Principle Component Analysis (and its use in Computer Vision) Wavelength 2 Wavelength 2 Principal Components Principal components are all about the directions
More information(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.
1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III
More informationLinear Algebra Review
Chapter 1 Linear Algebra Review It is assumed that you have had a course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc. I will review some of these terms here, but quite
More informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationMath Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationGeneralized Eigenvectors and Jordan Form
Generalized Eigenvectors and Jordan Form We have seen that an n n matrix A is diagonalizable precisely when the dimensions of its eigenspaces sum to n. So if A is not diagonalizable, there is at least
More informationChemometrics. Matti Hotokka Physical chemistry Åbo Akademi University
Chemometrics Matti Hotokka Physical chemistry Åbo Akademi University Linear regression Experiment Consider spectrophotometry as an example Beer-Lamberts law: A = cå Experiment Make three known references
More informationPrincipal Components Theory Notes
Principal Components Theory Notes Charles J. Geyer August 29, 2007 1 Introduction These are class notes for Stat 5601 (nonparametrics) taught at the University of Minnesota, Spring 2006. This not a theory
More informationand 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 informationAPPLICATIONS The eigenvalues are λ = 5, 5. An orthonormal basis of eigenvectors consists of
CHAPTER III APPLICATIONS The eigenvalues are λ =, An orthonormal basis of eigenvectors consists of, The eigenvalues are λ =, A basis of eigenvectors consists of, 4 which are not perpendicular However,
More informationJordan Normal Form and Singular Decomposition
University of Debrecen Diagonalization and eigenvalues Diagonalization We have seen that if A is an n n square matrix, then A is diagonalizable if and only if for all λ eigenvalues of A we have dim(u λ
More informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationEigenvalues and Eigenvectors 7.2 Diagonalization
Eigenvalues and Eigenvectors 7.2 Diagonalization November 8 Goals Suppose A is square matrix of order n. Provide necessary and sufficient condition when there is an invertible matrix P such that P 1 AP
More informationUsually, when we first formulate a problem in mathematics, we use the most familiar
Change of basis Usually, when we first formulate a problem in mathematics, we use the most familiar coordinates. In R, this means using the Cartesian coordinates x, y, and z. In vector terms, this is equivalent
More information