Eigenvalues and Eigenvectors

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Eigenvalues and Eigenvectors"

Transcription

1 Eigenvalues and Eigenvectors Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

2 Introduction We define eigenvalues and eigenvectors. We discuss how to compute them. We present their main properties. We finish with two applications, each could be a final project. Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

3 Definitions Definition A scalar λ is called an eigenvalue or a characteristic value of A if there is a nontrivial solution x of Ax = λx. Such a vector x is called an eigenvector of A corresponding to the eigenvalue λ. Let us note a few things: 1 Note that the zero vector x is always a solution of Ax = λx, it is why we are looking for nontrivial vectors x. 2 An eigenvalue can be 0, but not an eigenvector. Definition The eigenspace of A corresponding to the eigenvalue λ is the set of all eigenvectors of A corresponding to λ. Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

4 Computation Recall, we are trying to solve Ax = λx. Two things can happen. Ax = λx Ax λx = 0 Ax λi x = 0 (A λi ) x = 0 1 (A λi ) is invertible that is det (A λi ) 0. The only solution is the trivial solution. This is not of interest to us since we are seeking a nontrivial solution. 2 (A λi ) is not invertible that is det (A λi ) = 0. This is the only case in which we can hope to find a solution. Definition det (A λi ) is a polynomial of degree n. It is called the characteristic polynomial. Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

5 Computation 1 det (A λi ) is a polynomial of degree n, hence it has n roots, real and/or complex. Some of which may be repeated. 2 This means that if A is an n n matrix then it will have n eigenvalues, call them λ 1, λ 2,..., λ n. Some may be repeated. 3 If an eigenvalue λ appears only once in the list, it is called simple. 4 If an eigenvalue λ appears k > 1 times in the list, we say that λ has multiplicity k. 5 If λ 1, λ 2,..., λ k (k n) are the simple eigenvalues in the list, with corresponding eigenvectors x (1), x (2),.., x (k), then the eigenvectors are linearly independent. 6 If λ is an eigenvalue with multiplicity k > 1 then λ will have anywhere from 1 to k linearly independent eigenvectors. 7 If x is an eigenvector corresponding to λ then kx is also an eigenvector corresponding to λ. This means that eigenvectors are defined up to a constant. Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

6 Computation Example Find the eigenvalues and eigenvectors of A = Example Find the eigenvalues and eigenvectors of A = ( ) Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

7 Eigenvalues and MATLAB The MATLAB function to get the eigenvalues of a matrix is eig. It can be used different ways; we only show a few here. For a complete list, type help eig within MATLAB. 1 Given an n n matrix A, eig (A) will display the eigenvalues of A. Each eigenvalue will be printed as many times as its multiplicity. 2 Given an n n matrix A, s = eig (A) will find the eigenvalues of A and store them into the n 1 vector s. As above, each eigenvalue will appear as many times as its multiplicity. 3 Given an n n matrix A, [V D] = eig (A) will find the eigenvalues and eigenvectors of A. The eigenvectors of A will be stored in V as column vectors. So, V is in fact a matrix. The eigenvalues of A will be stored on the diagonal of D, the remaining entries of D being zeros. The eigenvalues will appear in the same order as the eigenvectors. Note that MATLAB will find eigenvectors which are unit vectors (magnitude 1). Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

8 Eigenvalues and MATLAB Example Test the eig function with A = Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

9 Properties of Eigenvalues and Eigenvectors We list some important properties. Their proof can be found in any linear algebra book. It will be useful to remember some properties of determinants. det ( A 1) 1 = det (A) det (AB) = det (A) det (B) det ( A ) T = det (A) det (ca) = c n det (A) Powers of a matrix: If Ax = λx then A 2 x = A (Ax) = A (λx) = λax = λ 2 x. In general, A n x = λ n x Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

10 Properties of Eigenvalues and Eigenvectors Eigenvalues of a triangular or diagonal matrix. Remembering that the determinant of a triangular or a diagonal matrix is the product of its entries on the diagonal, we see that the characteristic polynomial n of such a matrix is (a ii λ) hence the eigenvalues are a ii for i = 1..n. i=1 Similar Matrices: Recall that two matrices A and B are similar if there exists an invertible matrix P of the same size such that B = PAP 1. Two similar matrices have the same eigenvalues. λ is an eigenvalue of A if and only if 1 λ is an eigenvalue of A 1. If λ = a + bi is an eigenvalue of A with eigenvector v then λ = a bi is also an eigenvalue of A and its corresponding eigenvector is the conjugate of v. Symmetric Matrices always have real eigenvalues. Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

11 Properties of Eigenvalues and Eigenvectors Diagonalization. Suppose that v 1, v 2,..., v n are linearly independent vectors and λ 1, λ 2,..., λ n are their corresponding eigenvalues. Define P = [v 1 v 2... v n ] (note this is an n n matrix) and λ D = 0 λ then AP = PD that is A = PDP 1 or 0 0 λ n D = P 1 AP which means that the eigenvalues of A and D are the same. We can make the following important conclusions: If P is as defined, then P 1 AP is a diagonal matrix. det D = det ( P 1 AP ) = det ( P 1) det (A) det (P) = det (A) hence the determinant of a matrix is the product of its eigenvalues. Similarly, the trace of a matrix is the sum of its eigenvalues. The above implies that a matrix is invertible if and only if none of its eigenvalues is zero. Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

12 Properties of Eigenvalues and Eigenvectors Powers Matrix Revisited: If A = PDP 1 then A 2 = PDP 1 PDP 1 = PD 2 P 1. Similarly, A n = PD n P 1. Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

13 Applications An important application of eigenvalues and eigenvectors is with solving systems of first order differential equations. Google s page ranking algorithm uses a lot of linear algebra, including eigenvalues and eigenvectors. Here is a paper by Bryan and Leise on Google s PageRank algorithm. Eigenvalues for face recognition (eigenfaces). Here is the paper which started it all by Turk and Pentland. Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

14 Assignment 1 In the last section of this document, read and understand the paper on Google page ranking. This could be a potential project. 2 In the last section of this document, read and understand the paper on eigenfaces. This could be a potential project. Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall / 14

3.3 Eigenvalues and Eigenvectors

3.3 Eigenvalues and Eigenvectors .. EIGENVALUES AND EIGENVECTORS 27. Eigenvalues and Eigenvectors In this section, we assume A is an n n matrix and x is an n vector... Definitions In general, the product Ax results is another n vector

More information

Recall : Eigenvalues and Eigenvectors

Recall : 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 information

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero. Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial

More information

Study Guide for Linear Algebra Exam 2

Study 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 information

MAT 1302B Mathematical Methods II

MAT 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 information

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero. Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called

More information

Math 3191 Applied Linear Algebra

Math 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 information

Diagonalization of Matrix

Diagonalization 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 information

Lecture 15, 16: Diagonalization

Lecture 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 information

MAC Module 12 Eigenvalues and Eigenvectors. Learning Objectives. Upon completing this module, you should be able to:

MAC 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 information

MAC Module 12 Eigenvalues and Eigenvectors

MAC 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 information

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix

DIAGONALIZATION. 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 information

Math 2331 Linear Algebra

Math 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 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

Econ Slides from Lecture 7

Econ 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 information

ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors

ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 34 The powers of matrix Consider the following dynamic

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

More information

(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.

(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 information

Eigenvalues and Eigenvectors

Eigenvalues 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 information

Chapter 5 Eigenvalues and Eigenvectors

Chapter 5 Eigenvalues and Eigenvectors Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n

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

Dimension. Eigenvalue and eigenvector

Dimension. Eigenvalue and eigenvector Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,

More information

AMS10 HW7 Solutions. All credit is given for effort. (-5 pts for any missing sections) Problem 1 (20 pts) Consider the following matrix 2 A =

AMS10 HW7 Solutions. All credit is given for effort. (-5 pts for any missing sections) Problem 1 (20 pts) Consider the following matrix 2 A = AMS1 HW Solutions All credit is given for effort. (- pts for any missing sections) Problem 1 ( pts) Consider the following matrix 1 1 9 a. Calculate the eigenvalues of A. Eigenvalues are 1 1.1, 9.81,.1

More information

Announcements Wednesday, November 01

Announcements 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 information

The Singular Value Decomposition

The Singular Value Decomposition The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will

More information

(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).

(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a). .(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)

More information

Lecture 3 Eigenvalues and Eigenvectors

Lecture 3 Eigenvalues and Eigenvectors Lecture 3 Eigenvalues and Eigenvectors Eivind Eriksen BI Norwegian School of Management Department of Economics September 10, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 3 Eigenvalues and Eigenvectors

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

More information

Summer Session Practice Final Exam

Summer Session Practice Final Exam Math 2F Summer Session 25 Practice Final Exam Time Limit: Hours Name (Print): Teaching Assistant This exam contains pages (including this cover page) and 9 problems. Check to see if any pages are missing.

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

Announcements Monday, November 13

Announcements Monday, November 13 Announcements Monday, November 13 The third midterm is on this Friday, November 17 The exam covers 31, 32, 51, 52, 53, and 55 About half the problems will be conceptual, and the other half computational

More information

Lecture Notes: Eigenvalues and Eigenvectors. 1 Definitions. 2 Finding All Eigenvalues

Lecture Notes: Eigenvalues and Eigenvectors. 1 Definitions. 2 Finding All Eigenvalues Lecture Notes: Eigenvalues and Eigenvectors Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk 1 Definitions Let A be an n n matrix. If there

More information

Math Matrix Algebra

Math Matrix Algebra Math 44 - Matrix Algebra Review notes - 4 (Alberto Bressan, Spring 27) Review of complex numbers In this chapter we shall need to work with complex numbers z C These can be written in the form z = a+ib,

More information

Diagonalization. Hung-yi Lee

Diagonalization. Hung-yi Lee Diagonalization Hung-yi Lee Review If Av = λv (v is a vector, λ is a scalar) v is an eigenvector of A excluding zero vector λ is an eigenvalue of A that corresponds to v Eigenvectors corresponding to λ

More information

MAT1302F Mathematical Methods II Lecture 19

MAT1302F 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 information

1. In this problem, if the statement is always true, circle T; otherwise, circle F.

1. 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 information

City Suburbs. : population distribution after m years

City Suburbs. : population distribution after m years Section 5.3 Diagonalization of Matrices Definition Example: stochastic matrix To City Suburbs From City Suburbs.85.03 = A.15.97 City.15.85 Suburbs.97.03 probability matrix of a sample person s residence

More information

1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det

1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix

More information

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

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

More information

TMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013

TMA 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 information

The Jordan Normal Form and its Applications

The 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 information

Announcements Wednesday, November 01

Announcements 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 information

Linear Algebra: Matrix Eigenvalue Problems

Linear Algebra: Matrix Eigenvalue Problems CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

More information

Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI?

Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Section 5. The Characteristic Polynomial Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Property The eigenvalues

More information

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued) 1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix

More information

DM554 Linear and Integer Programming. Lecture 9. Diagonalization. Marco Chiarandini

DM554 Linear and Integer Programming. Lecture 9. Diagonalization. Marco Chiarandini DM554 Linear and Integer Programming Lecture 9 Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. More on 2. 3. 2 Resume Linear transformations and

More information

Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015

Chapters 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 information

Chapter 3. Determinants and Eigenvalues

Chapter 3. Determinants and Eigenvalues Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory

More information

Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation

More information

(the matrix with b 1 and b 2 as columns). If x is a vector in R 2, then its coordinate vector [x] B relative to B satisfies the formula.

(the matrix with b 1 and b 2 as columns). If x is a vector in R 2, then its coordinate vector [x] B relative to B satisfies the formula. 4 Diagonalization 4 Change of basis Let B (b,b ) be an ordered basis for R and let B b b (the matrix with b and b as columns) If x is a vector in R, then its coordinate vector x B relative to B satisfies

More information

Announcements Monday, November 06

Announcements Monday, November 06 Announcements Monday, November 06 This week s quiz: covers Sections 5 and 52 Midterm 3, on November 7th (next Friday) Exam covers: Sections 3,32,5,52,53 and 55 Section 53 Diagonalization Motivation: Difference

More information

Computationally, diagonal matrices are the easiest to work with. With this idea in mind, we introduce similarity:

Computationally, 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 information

LU Factorization. A m x n matrix A admits an LU factorization if it can be written in the form of A = LU

LU Factorization. A m x n matrix A admits an LU factorization if it can be written in the form of A = LU LU Factorization A m n matri A admits an LU factorization if it can be written in the form of Where, A = LU L : is a m m lower triangular matri with s on the diagonal. The matri L is invertible and is

More information

Diagonalization. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics

Diagonalization. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics Diagonalization MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Motivation Today we consider two fundamental questions: Given an n n matrix A, does there exist a basis

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

CS 246 Review of Linear Algebra 01/17/19

CS 246 Review of Linear Algebra 01/17/19 1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector

More information

ft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST

ft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST me me ft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST 1. (1 pt) local/library/ui/eigentf.pg A is n n an matrices.. There are an infinite number

More information

Announcements Monday, November 13

Announcements Monday, November 13 Announcements Monday, November 13 The third midterm is on this Friday, November 17. The exam covers 3.1, 3.2, 5.1, 5.2, 5.3, and 5.5. About half the problems will be conceptual, and the other half computational.

More information

Solving Linear Systems

Solving Linear Systems Solving Linear Systems Iterative Solutions Methods Philippe B. Laval KSU Fall 207 Philippe B. Laval (KSU) Linear Systems Fall 207 / 2 Introduction We continue looking how to solve linear systems of the

More information

Chapter 5. Eigenvalues and Eigenvectors

Chapter 5. Eigenvalues and Eigenvectors Chapter 5 Eigenvalues and Eigenvectors Section 5. Eigenvectors and Eigenvalues Motivation: Difference equations A Biology Question How to predict a population of rabbits with given dynamics:. half of the

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

What is on this week. 1 Vector spaces (continued) 1.1 Null space and Column Space of a matrix

What is on this week. 1 Vector spaces (continued) 1.1 Null space and Column Space of a matrix Professor Joana Amorim, jamorim@bu.edu What is on this week Vector spaces (continued). Null space and Column Space of a matrix............................. Null Space...........................................2

More information

Therefore, A and B have the same characteristic polynomial and hence, the same eigenvalues.

Therefore, A and B have the same characteristic polynomial and hence, the same eigenvalues. Similar Matrices and Diagonalization Page 1 Theorem If A and B are n n matrices, which are similar, then they have the same characteristic equation and hence the same eigenvalues. Proof Let A and B be

More information

4. Linear transformations as a vector space 17

4. 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 information

Chapter 4 & 5: Vector Spaces & Linear Transformations

Chapter 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 information

Math 205, Summer I, Week 4b:

Math 205, Summer I, Week 4b: Math 205, Summer I, 2016 Week 4b: Chapter 5, Sections 6, 7 and 8 (5.5 is NOT on the syllabus) 5.6 Eigenvalues and Eigenvectors 5.7 Eigenspaces, nondefective matrices 5.8 Diagonalization [*** See next slide

More information

Eigenvalues and Eigenvectors. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Eigenvalues and Eigenvectors. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Eigenvalues and Eigenvectors Consider the equation A x = λ x, where A is an nxn matrix. We call x (must be non-zero) an eigenvector of A if this equation can be solved for some value of λ. We call λ an

More information

Final Exam Practice Problems Answers Math 24 Winter 2012

Final 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 information

MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)

More information

Eigenvalues for Triangular Matrices. ENGI 7825: Linear Algebra Review Finding Eigenvalues and Diagonalization

Eigenvalues for Triangular Matrices. ENGI 7825: Linear Algebra Review Finding Eigenvalues and Diagonalization Eigenvalues for Triangular Matrices ENGI 78: Linear Algebra Review Finding Eigenvalues and Diagonalization Adapted from Notes Developed by Martin Scharlemann The eigenvalues for a triangular matrix are

More information

AN ITERATION. In part as motivation, we consider an iteration method for solving a system of linear equations which has the form x Ax = b

AN ITERATION. In part as motivation, we consider an iteration method for solving a system of linear equations which has the form x Ax = b AN ITERATION In part as motivation, we consider an iteration method for solving a system of linear equations which has the form x Ax = b In this, A is an n n matrix and b R n.systemsof this form arise

More information

Eigenvalues and Eigenvectors 7.1 Eigenvalues and Eigenvecto

Eigenvalues and Eigenvectors 7.1 Eigenvalues and Eigenvecto 7.1 November 6 7.1 Eigenvalues and Eigenvecto Goals Suppose A is square matrix of order n. Eigenvalues of A will be defined. Eigenvectors of A, corresponding to each eigenvalue, will be defined. Eigenspaces

More information

MA 265 FINAL EXAM Fall 2012

MA 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 information

Eigenvalues, Eigenvectors, and Diagonalization

Eigenvalues, 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 information

5.3.5 The eigenvalues are 3, 2, 3 (i.e., the diagonal entries of D) with corresponding eigenvalues. Null(A 3I) = Null( ), 0 0

5.3.5 The eigenvalues are 3, 2, 3 (i.e., the diagonal entries of D) with corresponding eigenvalues. Null(A 3I) = Null( ), 0 0 535 The eigenvalues are 3,, 3 (ie, the diagonal entries of D) with corresponding eigenvalues,, 538 The matrix is upper triangular so the eigenvalues are simply the diagonal entries, namely 3, 3 The corresponding

More information

Linear Algebra Primer

Linear Algebra Primer Linear Algebra Primer D.S. Stutts November 8, 995 Introduction This primer was written to provide a brief overview of the main concepts and methods in elementary linear algebra. It was not intended to

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

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

Linear Algebra. Rekha Santhanam. April 3, Johns Hopkins Univ. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, / 7

Linear 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 information

What is A + B? What is A B? What is AB? What is BA? What is A 2? and B = QUESTION 2. What is the reduced row echelon matrix of A =

What is A + B? What is A B? What is AB? What is BA? What is A 2? and B = QUESTION 2. What is the reduced row echelon matrix of A = STUDENT S COMPANIONS IN BASIC MATH: THE ELEVENTH Matrix Reloaded by Block Buster Presumably you know the first part of matrix story, including its basic operations (addition and multiplication) and row

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

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

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0. Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the

More information

Lecture 12: Diagonalization

Lecture 12: Diagonalization Lecture : Diagonalization A square matrix D is called diagonal if all but diagonal entries are zero: a a D a n 5 n n. () Diagonal matrices are the simplest matrices that are basically equivalent to vectors

More information

ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST

ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following

More information

Warm-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? 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 information

Review problems for MA 54, Fall 2004.

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

More information

First 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

First 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 information

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017

Math 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 information

Review Notes for Linear Algebra True or False Last Updated: January 25, 2010

Review Notes for Linear Algebra True or False Last Updated: January 25, 2010 Review Notes for Linear Algebra True or False Last Updated: January 25, 2010 Chapter 3 [ Eigenvalues and Eigenvectors ] 31 If A is an n n matrix, then A can have at most n eigenvalues The characteristic

More information

Eigenvalues for Triangular Matrices. ENGI 7825: Linear Algebra Review Finding Eigenvalues and Diagonalization

Eigenvalues for Triangular Matrices. ENGI 7825: Linear Algebra Review Finding Eigenvalues and Diagonalization Eigenvalues for Triangular Matrices ENGI 78: Linear Algebra Review Finding Eigenvalues and Diagonalization Adapted from Notes Developed by Martin Scharlemann June 7, 04 The eigenvalues for a triangular

More information

MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization.

MATH 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 information

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For

More information

Problems for M 10/26:

Problems for M 10/26: Math, Lesieutre Problem set # November 4, 25 Problems for M /26: 5 Is λ 2 an eigenvalue of 2? 8 Why or why not? 2 A 2I The determinant is, which means that A 2I has 6 a nullspace, and so there is an eigenvector

More information

LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for

More information

Math Camp Notes: Linear Algebra II

Math Camp Notes: Linear Algebra II Math Camp Notes: Linear Algebra II Eigenvalues Let A be a square matrix. An eigenvalue is a number λ which when subtracted from the diagonal elements of the matrix A creates a singular matrix. In other

More information

Chapter 2 Notes, Linear Algebra 5e Lay

Chapter 2 Notes, Linear Algebra 5e Lay Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication

More information

Eigenvalue and Eigenvector Homework

Eigenvalue and Eigenvector Homework Eigenvalue and Eigenvector Homework Olena Bormashenko November 4, 2 For each of the matrices A below, do the following:. Find the characteristic polynomial of A, and use it to find all the eigenvalues

More information