Properties of Matrices and Operations on Matrices


 Vivian Robbins
 1 years ago
 Views:
Transcription
1 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, and columns represent the variables or features that are observed for each unit. If the elements of a matrix X represent numeric observations on variables in the structure of a rectangular array as indicated above, the mathematical properties of X carry useful information about the observations and about the variables themselves. In addition, mathematical operations on the matrix may be useful in discovering structure in the data. These operations include various transformations and factorizations. 1
2 Symmetric Matrices A matrix A with elements a ij is said to be symmetric if each element a ji has the same value as a ij. Symmetric matrices have useful properties that we will mention from time to time. Symmetric matrices provide a generalization of the inner product. If A is symmetric and x and y are conformable vectors, then the bilinear form x T Ay has the property that x T Ay = y T Ax, and hence this operation on x and y is commutative, which is one of the properties of an inner product. More generally, a bilinear form is a kernel function of the two vectors, and a symmetric matrix corresponds to a symmetric kernel. An important type of bilinear form is x T Ax, which is called a quadratic form. 2
3 Nonnegative Definite and Positive Definite Matrices A real symmetric matrix A such that for any real conformable vector x the quadratic form x T Ax is nonnegative, that is, such that x T Ax 0, is called a nonnegative definite matrix. We denote the fact that A is nonnegative definite by A 0. (Note that we consider the zero matrix, 0 n n, to be nonnegative definite.) If the quadratic form is strictly positive, A is called a positive definite matrix and we write A 0. 3
4 Systems of Linear Equations One of the most common uses of matrices is to represent a system of linear equations Ax = b. Whether or not the system has a solution (that is, whether or not for a given A and b there is an x such that Ax = b) depends on the number of linearly independent rows in A (that is, considering each row of A as being a vector). The number of linearly independent rows of a matrix, which is also the number of linearly independent columns of the matrix, is called the rank of the matrix. A matrix is said to be of full rank if its rank is equal to either its number of rows or its number of columns. 4
5 A square full rank matrix is called a nonsingular matrix. We call a matrix that is square but not full rank singular. The system Ax = b has a solution if and only if rank(a b) rank(a), where A b is the matrix formed from A by adjoining b as an additional column. If a solution exists, the system is said to be consistent. The common regression equations do not satisfy the condition. 5
6 Matrix Inverses If the system Ax = b is consistent then x = A b is a solution, where A is any matrix such that AA A = A, as we can see by substituting A b into AA Ax = Ax. Given a matrix A, a matrix A such that AA A = A is called a generalized inverse of A, and we denote it as indicated. If A is square and of full rank, the generalized inverse, which is unique, is called the inverse and is denoted by A 1. It has a stronger property: AA 1 = A 1 A = I, where I is the identity matrix. 6
7 To the general requirement AA A = A, we successively add three requirements that define special generalized inverses, sometimes called respectively g 2, g 3, and g 4 inverses. The general generalized inverse is sometimes called a g 1 inverse. The g 4 inverse is called the MoorePenrose inverse. For a matrix A, a MoorePenrose inverse, denoted by A +, is a matrix that has four properties. 7
8 1. AA + A = A. Any matrix that satisfies this condition is called a generalized inverse, and as we have seen above is denoted by A. For many applications, this is the only condition necessary. Such a matrix is also called a g 1 inverse, an inner pseudoinverse, or a conditional inverse. 2. A + AA + = A +. A matrix A + that satisfies this condition is called an outer pseudoinverse. A g 1 inverse that also satisfies this condition is called a g 2 inverse or reflexive generalized inverse, and is denoted by A. 3. A + A is symmetric. 4. AA + is symmetric. 8
9 The Matrix X T X When numerical data are stored in the usual way in a matrix X, the matrix X T X often plays an important role in statistical analysis. A matrix of this form is called a Gramian matrix, and it has some interesting properties. First of all, we note that X T X is symmetric; that is, the (ij) th element, k x k,i x k,j is the same as the (ji) th element. Secondly, because for any y, (Xy) T Xy 0, X T X is nonnegative definite. Next we note that X T X = 0 X = 0. 9
10 The generalized inverses of X T X have useful properties. First, we see from the definition, for any generalized inverse (X T X), that ((X T X) ) T is also a generalized inverse of X T X. (Note that (X T X) is not necessarily symmetric.) Also, we have X(X T X) X T X = X. This means that (X T X) X T is a generalized inverse of X. The MoorePenrose inverse of X has an interesting relationship with a generalized inverse of X T X: XX + = X(X T X) X T. 10
11 An important property of X(X T X) X T is its invariance to the choice of the generalized inverse of X T X. The matrix X(X T X) X T has a number of other interesting properties in addition to those mentioned above. ( X(X T X) X T) ( X(X T X) X T) = X(X T X) (X T X)(X T X) X T that is, X(X T X) X T is idempotent. = X(X T X) X T, It is clear that the only idempotent matrix that is of full rank is the identity I. 11
12 Any real symmetric idempotent matrix is a projection matrix. The most familiar application of the matrix X(X T X) X T is in the analysis of the linear regression model y = Xβ + ɛ. This matrix projects the observed vector y onto a lowerdimensional subspace that represents the fitted model: ŷ = X(X T X) X T y. Projection matrices, as the name implies, generally transform or project a vector onto a lowerdimensional subspace. 12
13 Eigenvalues and Eigenvectors Multiplication of a given vector by a square matrix may result in a scalar multiple of the vector. If A is an n n matrix, v is a vector not equal to 0, and c is a scalar such that Av = cv, we say v is an eigenvector of A and c is an eigenvalue of A. We should note how remarkable the relationship Av = cv is: The effect of a matrix multiplication of an eigenvector is the same as a scalar multiplication of the eigenvector. The eigenvector is an invariant of the transformation in the sense that its direction does not change under the matrix multiplication transformation. 13
14 Eigenvalues and Eigenvectors We immediately see that if an eigenvalue of a matrix A is 0, then A must be singular. We also note that if v is an eigenvector of A, and t is any nonzero scalar, tv is also an eigenvector of A. Hence, we can normalize eigenvectors, and we often do. If A is symmetric there are several useful facts about its eigenvalues and eigenvectors. The eigenvalues and eigenvector of a (real) symmetric matrix are all real. 14
15 Eigenvalues and Eigenvectors The eigenvectors of a symmetric matrix are (or can be chosen to be) mutually orthogonal. We can therefore represent a symmetric matrix A as A = V CV T, where V is an orthogonal matrix whose columns are the eigenvectors of A and C is a diagonal matrix whose (ii) th element is the eigenvalue corresponding to the eigenvector in the i th column of V. This is called the diagonal factorization of A. 15
16 Eigenvalues and Eigenvectors If A is a nonnegative (positive) definite matrix, and c is an eigenvalue with corresponding eigenvector v, if we multiply both sides of the equation Av = cv, we have v T Av = cv T v 0(> 0), and since v T v > 0, we have c 0(> 0). The maximum modulus of any eigenvalue in a given matrix is of interest. This value is called the spectral radius, and for the matrix A, is denoted by ρ(a): ρ(a) = max c i, where the c i s are the eigenvalues of A. The spectral radius is very important in many applications, from both computational and statistical standpoints. The convergence of some iterative algorithms, for example, depend on bounds on the spectral radius. 16
17 Matrix Decomposition Computations with matrices are often facilitated by first decomposing the matrix into multiplicative factors that are easier to work with computationally, or else reveal some important characteristics of the matrix. Some decompositions exist only for special types of matrices, such as symmetric matrices or positive definite matrices. 17
18 The Singular Value Decomposition One of most useful decompositions, and one that applies to all types of matrices, is the singular value decomposition. An n m matrix A can be factored as A = UDV T, where U is an n n orthogonal matrix, V is an m m orthogonal matrix, and D is an n m diagonal matrix with nonnegative entries. The number of positive entries in D is the same as the rank of A. This factorization is called the singular value decomposition (SVD) or the canonical singular value factorization of A. 18
19 Singular Values and the Singular Value Decomposition The elements on the diagonal of D, d i, are called the singular values of A. We can rearrange the entries in D so that d 1 d 2, and by rearranging the columns of U correspondingly, nothing is changed. If the rank of the matrix is r, we have d 1 d r > 0, and if r < min(n, m), then d r+1 = = d min(n,m) = 0. In this case D = where D r = diag(d 1,..., d r ). [ Dr ], From the factorization defining the singular values, we see that the singular values of A T are the same as those of A. 19
20 Singular Values and the Singular Value Decomposition For a matrix with more rows than columns, in an alternate definition of the singular value decomposition, the matrix U is n m with orthogonal columns, and D is an m m diagonal matrix with nonnegative entries. Likewise, for a matrix with more columns than rows, the singular value decomposition can be defined as above but with the matrix V being m n with orthogonal columns and D being m m and diagonal with nonnegative entries. If A is symmetric its singular values are the absolute values of its eigenvalues. 20
21 SVD and the MoorePenrose Inverse The MoorePenrose inverse of a matrix has a simple relationship to its SVD. If the SVD of A is given by UDV T, then its MoorePenrose inverse is A + = V D + U T, as is easy to verify. The MoorePenrose inverse of D is just the matrix D + formed by inverting all of the positive entries of D and leaving the other entries unchanged. 21
22 Square Root Factorization of a Nonnegative Definite Matrix If A is a nonnegative definite matrix (which, for me, means that it is symmetric), its eigenvalues are nonnegative, so we can write S = C 1 2, where S is a diagonal matrix whose elements are the square roots of the elements in the C matrix in the diagonal factorization of A. Now we observe that (V SV T ) 2 = V CV T = A; hence, we write and we have (A 1 2) 2 = A. A 1 2 = V SV T, 22
Chapter 3. Matrices. 3.1 Matrices
40 Chapter 3 Matrices 3.1 Matrices Definition 3.1 Matrix) A matrix A is a rectangular array of m n real numbers {a ij } written as a 11 a 12 a 1n a 21 a 22 a 2n A =.... a m1 a m2 a mn The array has m rows
More informationB553 Lecture 5: Matrix Algebra Review
B553 Lecture 5: Matrix Algebra Review Kris Hauser January 19, 2012 We have seen in prior lectures how vectors represent points in R n and gradients of functions. Matrices represent linear transformations
More informationPseudoinverse & MoorePenrose Conditions
ECE 275AB Lecture 7 Fall 2008 V1.0 c K. KreutzDelgado, UC San Diego p. 1/1 Lecture 7 ECE 275A Pseudoinverse & MoorePenrose Conditions ECE 275AB Lecture 7 Fall 2008 V1.0 c K. KreutzDelgado, UC San Diego
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationReview 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 informationLinear 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 informationProposition 42. Let M be an m n matrix. Then (32) N (M M)=N (M) (33) R(MM )=R(M)
RODICA D. COSTIN. Singular Value Decomposition.1. Rectangular matrices. For rectangular matrices M the notions of eigenvalue/vector cannot be defined. However, the products MM and/or M M (which are square,
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A pdimensional 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 informationMatrices and Vectors. Definition of Matrix. An MxN matrix A is a twodimensional array of numbers A =
30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a twodimensional 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 informationMaths for Signals and Systems Linear Algebra in Engineering
Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE
More informationLecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε,
2. REVIEW OF LINEAR ALGEBRA 1 Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε, where Y n 1 response vector and X n p is the model matrix (or design matrix ) with one row for
More informationStat 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 informationComputational math: Assignment 1
Computational math: Assignment 1 Thanks Ting Gao for her Latex file 11 Let B be a 4 4 matrix to which we apply the following operations: 1double column 1, halve row 3, 3add row 3 to row 1, 4interchange
More informationRecall the convention that, for us, all vectors are column vectors.
Some linear algebra Recall the convention that, for us, all vectors are column vectors. 1. Symmetric matrices Let A be a real matrix. Recall that a complex number λ is an eigenvalue of A if there exists
More informationReview of some mathematical tools
MATHEMATICAL FOUNDATIONS OF SIGNAL PROCESSING Fall 2016 Benjamín Béjar Haro, Mihailo Kolundžija, Reza Parhizkar, Adam Scholefield Teaching assistants: Golnoosh Elhami, Hanjie Pan Review of some mathematical
More informationMatrix Algebra, part 2
Matrix Algebra, part 2 MingChing Luoh 2005.9.12 1 / 38 Diagonalization and Spectral Decomposition of a Matrix Optimization 2 / 38 Diagonalization and Spectral Decomposition of a Matrix Also called Eigenvalues
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 informationNumerical Methods. Elena loli Piccolomini. Civil Engeneering. piccolom. Metodi Numerici M p. 1/??
Metodi Numerici M p. 1/?? Numerical Methods Elena loli Piccolomini Civil Engeneering http://www.dm.unibo.it/ piccolom elena.loli@unibo.it Metodi Numerici M p. 2/?? Least Squares Data Fitting Measurement
More information2. Review of Linear Algebra
2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear
More information2. Matrix Algebra and Random Vectors
2. Matrix Algebra and Random Vectors 2.1 Introduction Multivariate data can be conveniently display as array of numbers. In general, a rectangular array of numbers with, for instance, n rows and p columns
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More 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 informationImage Registration Lecture 2: Vectors and Matrices
Image Registration Lecture 2: Vectors and Matrices Prof. Charlene Tsai Lecture Overview Vectors Matrices Basics Orthogonal matrices Singular Value Decomposition (SVD) 2 1 Preliminary Comments Some of this
More informationSVD and its Application to Generalized Eigenvalue Problems. Thomas Melzer
SVD and its Application to Generalized Eigenvalue Problems Thomas Melzer June 8, 2004 Contents 0.1 Singular Value Decomposition.................. 2 0.1.1 Range and Nullspace................... 3 0.1.2
More information(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =
. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)
More informationEE731 Lecture Notes: Matrix Computations for Signal Processing
EE731 Lecture Notes: Matrix Computations for Signal Processing James P. Reilly c Department of Electrical and Computer Engineering McMaster University October 17, 005 Lecture 3 3 he Singular Value Decomposition
More informationComputational Methods. Eigenvalues and Singular Values
Computational Methods Eigenvalues and Singular Values Manfred Huber 2010 1 Eigenvalues and Singular Values Eigenvalues and singular values describe important aspects of transformations and of data relations
More informationLinear 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 informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationLinear 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 informationReal symmetric matrices/1. 1 Eigenvalues and eigenvectors
Real symmetric matrices 1 Eigenvalues and eigenvectors We use the convention that vectors are row vectors and matrices act on the right. Let A be a square matrix with entries in a field F; suppose that
More information5.6. PSEUDOINVERSES 101. A H w.
5.6. PSEUDOINVERSES 0 Corollary 5.6.4. If A is a matrix such that A H A is invertible, then the leastsquares solution to Av = w is v = A H A ) A H w. The matrix A H A ) A H is the left inverse of A and
More information1 Linearity and Linear Systems
Mathematical Tools for Neuroscience (NEU 34) Princeton University, Spring 26 Jonathan Pillow Lecture 78 notes: Linear systems & SVD Linearity and Linear Systems Linear system is a kind of mapping f( x)
More informationMathematical Methods wk 2: Linear Operators
John Magorrian, magog@thphysoxacuk These are workinprogress notes for the secondyear course on mathematical methods The most uptodate version is available from http://wwwthphysphysicsoxacuk/people/johnmagorrian/mm
More informationThe University of Texas at Austin Department of Electrical and Computer Engineering. EE381V: Large Scale Learning Spring 2013.
The University of Texas at Austin Department of Electrical and Computer Engineering EE381V: Large Scale Learning Spring 2013 Assignment Two Caramanis/Sanghavi Due: Tuesday, Feb. 19, 2013. Computational
More informationChapter Two Elements of Linear Algebra
Chapter Two Elements of Linear Algebra Previously, in chapter one, we have considered single first order differential equations involving a single unknown function. In the next chapter we will begin to
More informationPhys 201. Matrices and Determinants
Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1
More informationSection 3.3. Matrix Rank and the Inverse of a Full Rank Matrix
3.3. Matrix Rank and the Inverse of a Full Rank Matrix 1 Section 3.3. Matrix Rank and the Inverse of a Full Rank Matrix Note. The lengthy section (21 pages in the text) gives a thorough study of the rank
More informationMath 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook.
Math 443 Differential Geometry Spring 2013 Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Endomorphisms of a Vector Space This handout discusses
More information1. The Polar Decomposition
A PERSONAL INTERVIEW WITH THE SINGULAR VALUE DECOMPOSITION MATAN GAVISH Part. Theory. The Polar Decomposition In what follows, F denotes either R or C. The vector space F n is an inner product space with
More information(VII.E) The Singular Value Decomposition (SVD)
(VII.E) The Singular Value Decomposition (SVD) In this section we describe a generalization of the Spectral Theorem to nonnormal operators, and even to transformations between different vector spaces.
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationMatrices A brief introduction
Matrices A brief introduction Basilio Bona DAUIN Politecnico di Torino September 2013 Basilio Bona (DAUIN) Matrices September 2013 1 / 74 Definitions Definition A matrix is a set of N real or complex numbers
More informationSTAT 8260 Theory of Linear Models Lecture Notes
STAT 8260 Theory of Linear Models Lecture Notes Classical linear models are at the core of the field of statistics, and are probably the most commonly used set of statistical techniques in practice. For
More informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationDefinition (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 informationSolutions 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 informationNONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction
NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques
More informationVectors and Matrices Statistics with Vectors and Matrices
Vectors and Matrices Statistics with Vectors and Matrices Lecture 3 September 7, 005 Analysis Lecture #39/7/005 Slide 1 of 55 Today s Lecture Vectors and Matrices (Supplement A  augmented with SAS proc
More informationA 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 informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationLinear 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 informationApplied Mathematics 205. Unit II: Numerical Linear Algebra. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit II: Numerical Linear Algebra Lecturer: Dr. David Knezevic Unit II: Numerical Linear Algebra Chapter II.3: QR Factorization, SVD 2 / 66 QR Factorization 3 / 66 QR Factorization
More informationEE263: Introduction to Linear Dynamical Systems Review Session 5
EE263: Introduction to Linear Dynamical Systems Review Session 5 Outline eigenvalues and eigenvectors diagonalization matrix exponential EE263 RS5 1 Eigenvalues and eigenvectors we say that λ C is an eigenvalue
More information18.06 Problem Set 8  Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8  Solutions Due Wednesday, 4 November 2007 at 4 pm in 206 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
More informationChapter 3: Theory Review: Solutions Math 308 F Spring 2015
Chapter : Theory Review: Solutions Math 08 F Spring 05. What two properties must a function T : R m R n satisfy to be a linear transformation? (a) For all vectors u and v in R m, T (u + v) T (u) + T (v)
More informationLecture 6 Positive Definite Matrices
Linear Algebra Lecture 6 Positive Definite Matrices Prof. ChunHung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices
More informationContents. 1 Vectors, Lines and Planes 1. 2 Gaussian Elimination Matrices Vector Spaces and Subspaces 124
Matrices Math 220 Copyright 2016 Pinaki Das This document is freely redistributable under the terms of the GNU Free Documentation License For more information, visit http://wwwgnuorg/copyleft/fdlhtml Contents
More informationidentity matrix, shortened I the jth column of I; the jth standard basis vector matrix A with its elements a ij
Notation R R n m R n m r R n s real numbers set of n m real matrices subset of R n m consisting of matrices with rank r subset of R n n consisting of symmetric matrices NND n subset of R n s consisting
More informationLecture 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 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 informationLeast Squares Optimization
Least Squares Optimization The following is a brief review of least squares optimization and constrained optimization techniques. I assume the reader is familiar with basic linear algebra, including the
More informationLecture 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 informationMatrices 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 informationTopic 2 Quiz 2. choice C implies B and B implies C. correctchoice C implies B, but B does not imply C
Topic 1 Quiz 1 text A reduced rowechelon 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 correctchoice may have 0, 1, 2, or 3 choice may have 0,
More informationMath 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 informationThe Singular Value Decomposition
The Singular Value Decomposition An Important topic in NLA Radu Tiberiu Trîmbiţaş BabeşBolyai University February 23, 2009 Radu Tiberiu Trîmbiţaş ( BabeşBolyai University)The Singular Value Decomposition
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationSymmetric matrices and dot products
Symmetric matrices and dot products Proposition An n n matrix A is symmetric iff, for all x, y in R n, (Ax) y = x (Ay). Proof. If A is symmetric, then (Ax) y = x T A T y = x T Ay = x (Ay). If equality
More informationMATRIX ALGEBRA. or x = (x 1,..., x n ) R n. y 1 y 2. x 2. x m. y m. y = cos θ 1 = x 1 L x. sin θ 1 = x 2. cos θ 2 = y 1 L y.
as Basics Vectors MATRIX ALGEBRA An array of n real numbers x, x,, x n is called a vector and it is written x = x x n or x = x,, x n R n prime operation=transposing a column to a row Basic vector operations
More informationSupplementary Notes on Linear Algebra
Supplementary Notes on Linear Algebra Mariusz Wodzicki May 3, 2015 1 Vector spaces 1.1 Coordinatization of a vector space 1.1.1 Given a basis B = {b 1,..., b n } in a vector space V, any vector v V can
More informationChapter 0 Miscellaneous Preliminaries
EE 520: Topics Compressed Sensing Linear Algebra Review Notes scribed by Kevin Palmowski, Spring 2013, for Namrata Vaswani s course Notes on matrix spark courtesy of Brian Lois More notes added by Namrata
More informationLinear 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 informationBasic 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 informationLinear 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 informationMatrices. Chapter What is a Matrix? We review the basic matrix operations. An array of numbers a a 1n A = a m1...
Chapter Matrices We review the basic matrix operations What is a Matrix? An array of numbers a a n A = a m a mn with m rows and n columns is a m n matrix Element a ij in located in position (i, j The elements
More informationDSGA 1002 Lecture notes 10 November 23, Linear models
DSGA 2 Lecture notes November 23, 2 Linear functions Linear models A linear model encodes the assumption that two quantities are linearly related. Mathematically, this is characterized using linear functions.
More informationTHE SINGULAR VALUE DECOMPOSITION MARKUS GRASMAIR
THE SINGULAR VALUE DECOMPOSITION MARKUS GRASMAIR 1. Definition Existence Theorem 1. Assume that A R m n. Then there exist orthogonal matrices U R m m V R n n, values σ 1 σ 2... σ p 0 with p = min{m, n},
More informationQuadratic forms. Here. Thus symmetric matrices are diagonalizable, and the diagonalization can be performed by means of an orthogonal matrix.
Quadratic forms 1. Symmetric matrices An n n matrix (a ij ) n ij=1 with entries on R is called symmetric if A T, that is, if a ij = a ji for all 1 i, j n. We denote by S n (R) the set of all n n symmetric
More information10725/36725: Convex Optimization Prerequisite Topics
10725/36725: Convex Optimization Prerequisite Topics February 3, 2015 This is meant to be a brief, informal refresher of some topics that will form building blocks in this course. The content of the
More informationConjugate Gradient (CG) Method
Conjugate Gradient (CG) Method by K. Ozawa 1 Introduction In the series of this lecture, I will introduce the conjugate gradient method, which solves efficiently large scale sparse linear simultaneous
More informationSingular Value Decomposition
Chapter 6 Singular Value Decomposition In Chapter 5, we derived a number of algorithms for computing the eigenvalues and eigenvectors of matrices A R n n. Having developed this machinery, we complete our
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationHomework 1 Elena Davidson (B) (C) (D) (E) (F) (G) (H) (I)
CS 106 Spring 2004 Homework 1 Elena Davidson 8 April 2004 Problem 1.1 Let B be a 4 4 matrix to which we apply the following operations: 1. double column 1, 2. halve row 3, 3. add row 3 to row 1, 4. interchange
More informationSingular Value Decomposition
Singular Value Decomposition Motivatation The diagonalization theorem play a part in many interesting applications. Unfortunately not all matrices can be factored as A = PDP However a factorization A =
More informationReview of Vectors and Matrices
A P P E N D I X D Review of Vectors and Matrices D. VECTORS D.. Definition of a Vector Let p, p, Á, p n be any n real numbers and P an ordered set of these real numbers that is, P = p, p, Á, p n Then P
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationElementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th.
Elementary Linear Algebra Review for Exam Exam is Monday, November 6th. The exam will cover sections:.4,..4, 5. 5., 7., the class notes on Markov Models. You must be able to do each of the following. Section.4
More information1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )
Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems PerOlof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical
More informationSymmetric Matrices and Eigendecomposition
Symmetric Matrices and Eigendecomposition Robert M. Freund January, 2014 c 2014 Massachusetts Institute of Technology. All rights reserved. 1 2 1 Symmetric Matrices and Convexity of Quadratic Functions
More information1 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 informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Regularization: Ridge Regression and Lasso Week 14, Lecture 2
MA 575 Linear Models: Cedric E. Ginestet, Boston University Regularization: Ridge Regression and Lasso Week 14, Lecture 2 1 Ridge Regression Ridge regression and the Lasso are two forms of regularized
More informationInverse of a Square Matrix. For an N N square matrix A, the inverse of A, 1
Inverse of a Square Matrix For an N N square matrix A, the inverse of A, 1 A, exists if and only if A is of full rank, i.e., if and only if no column of A is a linear combination 1 of the others. A is
More informationPositive Definite Matrix
1/29 ChiaPing Chen Professor Department of Computer Science and Engineering National Sun Yatsen University Linear Algebra Positive Definite, Negative Definite, Indefinite 2/29 Pure Quadratic Function
More informationMatrix & Linear Algebra
Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A
More informationLecture 3: QRFactorization
Lecture 3: QRFactorization This lecture introduces the Gram Schmidt orthonormalization process and the associated QRfactorization of matrices It also outlines some applications of this factorization
More informationLecture 5 Singular value decomposition
Lecture 5 Singular value decomposition Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University, tieli@pku.edu.cn
More informationThe Eigenvalue Problem: Perturbation Theory
Jim Lambers MAT 610 Summer Session 200910 Lecture 13 Notes These notes correspond to Sections 7.2 and 8.1 in the text. The Eigenvalue Problem: Perturbation Theory The Unsymmetric Eigenvalue Problem Just
More informationLinear Algebra in Actuarial Science: Slides to the lecture
Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a ToolBox Linear Equation Systems Discretization of differential equations: solving linear equations
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More information