ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors

Size: px
Start display at page:

Download "ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors"

Transcription

1 ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors Xiaohui Xie University of California, Irvine Xiaohui Xie (UCI) ICS 6N 1 / 34

2 The powers of matrix Consider the following dynamic system: x (t+1) = Ax (t) where A is an n n matrix and x (t) is vector in R n. How to compute x (100)? x (t+1) = A t x (1) Need to find ways to efficiently calculate A t. Xiaohui Xie (UCI) ICS 6N 2 / 34

3 Eigenvalues and eigenvectors An eigenvector of an n n matrix A is a nonzero vector x such that for some scalar λ. Ax = λx A scalar λ is called an eigenvalue of A if there is a nontrivial solution x of Ax = λx; such an x is called an eigenvector corresponding to λ. Xiaohui Xie (UCI) ICS 6N 3 / 34

4 Example A = [ ] 3 2, v = 1 0 [ ] 2 1 [ ] 4 Av = = 2v 2 So λ = 2 is an eigenvalue, and v is the corresponding eigenvector. Xiaohui Xie (UCI) ICS 6N 4 / 34

5 Finding eigenvalues Ax = λx Ax λx = 0 Ax λix = 0 (A λi )x = 0 So in order for x to be an eigenvector, x is a nontrivial solution to (A λi )x = 0 det(a λi ) = 0 Xiaohui Xie (UCI) ICS 6N 5 / 34

6 Eigenvalues of triangular matrices A is an upper triangular matrix if all values below diagonal are zero; lower triangular if all values above diagonal are zero. Determinant of triangular matrix is the product of diagonal entries. Eigenvalues of triangular matrix are diagonal entries. Xiaohui Xie (UCI) ICS 6N 6 / 34

7 The following statements are equivalent regarding nxn matrix A 1) Ax = 0 has nontrivial solutions 2) A is non invertible 3) det(a) = 0 4) Null(A) { 0 } 5) dim(null(a)) 1 6) Rank(A) < n 7) The column vectors are linearly dependent 8) dim(col(a)) < n Xiaohui Xie (UCI) ICS 6N 7 / 34

8 Finding eigenvalues Find the roots of the characteristic polynomial equation: det(a λi ) = 0 a 11 λ... a 1n A λi =... a n1... a nn λ Once an eigenvalue is discovered, find its corresponding eigenvector by solving (A λi )x = 0. Xiaohui Xie (UCI) ICS 6N 8 / 34

9 Example [ ] 3 2 A = 1 0 det(a λi ) = 3 λ 2 1 λ = λ(3 λ) + 2 = λ 2 3λ + 2 = 0 = (λ 2)(λ 1) = 0 λ = 1 or λ = 2 There is a respective eigenvector for each eigenvalue Xiaohui Xie (UCI) ICS 6N 9 / 34

10 For λ = 1 (A λi )x = [ ] 2 2 x = 0 = x = 1 1 [ ] 1 1 For λ = 2 (A λi )x = [ ] 1 2 x = 0 = x = 1 2 [ ] 2 1 Xiaohui Xie (UCI) ICS 6N 10 / 34

11 Eigenvector corresponding to eigenvalue λ Suppose we have found a λ with det(a λi ) = 0. Find its corresponding eigenvector by solving Ax = λx Any nonzero vector in Null(A λi ), called eigenspace corresponding to λ, is a corresponding eigenvector Xiaohui Xie (UCI) ICS 6N 11 / 34

12 Example A = For λ = 2, the augmented matrix corresponding to Ax = λx is Solutions: basic variables: x 1 ; free variables: x 2 and x 3. x 1 1/2 3 x = x 2 = x x 3 0 = x 2 v 1 + x 3 v 2 x Eigenspace: span{v 1, v 2 } Xiaohui Xie (UCI) ICS 6N 12 / 34

13 Characteristic polynomial For matrix A in the previous slide, the characteristic polynomial is of order 3 det(a λi ) = 0 (λ 2) 2 (λ c) = 0 In this case λ = 2 has a multiplicity of 2 In general, the characteristic polynomial of an n n matrix A is of order n det(a λi ) = 0 (λ λ 1 )(λ λ 2 )... (λ λ n ) = 0 The multiplicity of a root is the number of times the root appears in the characteristic polynomial decomposition. Xiaohui Xie (UCI) ICS 6N 13 / 34

14 Example A = λ The characteristic equation: 0 = 0 3 λ λ λ It is an upper triangular matrix. its the determinant is the product of the diagonal entries det(a λi ) = (λ 5) 2 (λ 1)(λ 3) : Multiplicity of λ = 5 is 2, while the rest is 1. Xiaohui Xie (UCI) ICS 6N 14 / 34

15 Dimension of eigenspace If the multiplicity of an eigenvalue is exactly one, then dim(null(a λi )) = 1, so there is only one eigenvector up to a scale difference. Given a eigenvalue λ, dim(null(a λi )) its multiplicity Xiaohui Xie (UCI) ICS 6N 15 / 34

16 When eigenvalue λ = 0 The following statements are equivalent: λ = 0 det(a) = 0 A is not invertible (also called singular) Xiaohui Xie (UCI) ICS 6N 16 / 34

17 Matrix diagonalization Suppose nxn matrix A has n linearly independent eigenvectors v 1, v 2,..., v n with corresponding eigenvalues λ 1, λ 2,..., λ n (some of these eigenvalues might be equal): Let P = [ v 1... v n ]. Then Av 1 = λ 1 v 1,, Av n = λ n v n AP = [ Av 1 Av 2... Av n ] = [ λ1 v 1 λ 2 v 2... λ n v n ] = PΛ λ with Λ = λ n A = PΛP 1. A is called diagonalizable if this is true. Xiaohui Xie (UCI) ICS 6N 17 / 34

18 Calculate the powers of matrix If A is diagonalizable, A n = AA... A = PΛP 1 PΛP 1... PΛP 1 = PΛ n P 1 λ n = P P λ n n Xiaohui Xie (UCI) ICS 6N 18 / 34

19 Solving dynamical systems Consider the following discrete dynamical system x t+1 = Ax t with the initial x 0. How to calculate x t? Solution (if A can be diagonalized): x t = A t x 0 = PΛ t P 1 x 0. Let c = P 1 x 0, that is Pc = x 0. Then x t = c 1 λ t 1v 1 + c 2 λ t 2v c n λ t nv n written as a linear combination of eigenvectors. Xiaohui Xie (UCI) ICS 6N 19 / 34

20 Suppose A has n linearly independent eigenvectors v 1, v 2,..., v n with corresponding eigenvalues λ 1, λ 2,..., λ n then A can be diagonalized as: A = PΛP 1 (diagonalization of A) where P = [ ] v 1 v 2... v n λ Λ = λ n And this is very useful to calculate the power of a matrix A k = PΛ k P 1 (λ 1 ) n 0 0 Λ k = P P (λ n ) n Xiaohui Xie (UCI) ICS 6N 20 / 34

21 Steps for matrix diagonalization Diagonalize the following matrix A = Find eigenvalues λ 1, λ 2, λ 3 Find three linearly independent eigenvectors of A Construct P = [v 1, v 2, v 3 ] Construct D Check AP = PD and A = PDP 1 Xiaohui Xie (UCI) ICS 6N 21 / 34

22 Not all matrices are diagonalizable An example of non-diagonalizable matrix A = Eigenvalues: λ 1 = λ 2 = λ 3 = 2. Has only one eigenvalue with multiplicity of 3. However, (A 2I )x = 0 = x 2 = 0, x 3 = 0 with x 1 free Thus A has only one eigenvector: 1 0, and cannot be diagonalized. 0 Xiaohui Xie (UCI) ICS 6N 22 / 34

23 Matrix with distinct eigenvalues If v 1,, v r are eigenvectors that correspond to distinct eigenvalues λ 1,, λ r of an n n matrix A, then the set {v 1,, v r } is linearly independent. An n n matrix with n distinct eigenvalues is diagonalizable. Xiaohui Xie (UCI) ICS 6N 23 / 34

24 Matrices whose eigenvalues are not distinct Let A be an n n matrix whose distinct eigenvalues are λ 1,, λ p. For 1 k p, the dimension of the eigenspace for λ k the multiplicity of λ k A is diagonalizable if and only if the sum of the dimensions of the eigenvalues equals n. Xiaohui Xie (UCI) ICS 6N 24 / 34

25 Application: discrete dynamical systems Let s study an ecological problem, in particular, a predator-prey system involving two species: owl and wood rat. Denote owl and wood rate populations at time t (unit: month) by [ ] Ot x t = where O t is the number of owls (in unit 1) in the region, and R t is the number of wood rats (in unit thousands) in the region. Suppose the two populations are modeled by R t O t+1 = 0.5O t + 0.4R t (1) R t+1 = 0.104O t + 1.1R t (2) Xiaohui Xie (UCI) ICS 6N 25 / 34

26 Application: predator-prey system Write the population dynamics in matrix format: x t+1 = Ax t with [ ] A = A has two eigenvalues λ 1 = 1.02 and λ 2 = 0.58 with corresponding eigenvectors [ ] [ ] 10 5 v 1 =, v 13 2 = 1 Let x 0 = c 1 v 1 + c 2 v 2. Then x t = c t v 1 + c t v 2 c t v 1 where the approximation is true when t is large. In another words, x t+1 = 1.02x t when t is large. Both species will grow 2% monthly. Every 10 owns, there are about 13 thousands rats. Xiaohui Xie (UCI) ICS 6N 26 / 34

27 Application of matrix diagonalization Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13 How to find the 100-th number x 100? Xiaohui Xie (UCI) ICS 6N 27 / 34

28 Application of matrix diagonalization Solution y k = Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13 How to find the 100-th number x 100 quickly? [ xk x k+1 then [ ] 0 y 0 = 1 So y k+1 = ] [ xk+1 x k+2 ] [ = x k+1 x k + x k+1 ] = [ ] [ ] 0 1 xk 1 1 x k+1 Then y k+1 = Ay k So we only have to calculate A to the desired power to solve this. This means we need the diagonalization of A Xiaohui Xie (UCI) ICS 6N 28 / 34

29 From det(a λi ) = λ 2 λ 1, we have λ 1 = = 1.618, λ 2 = The corresponding eigenvectors are [ ] [ ] 1 1 v 1 =, v 2 = λ 1 λ 2 = Xiaohui Xie (UCI) ICS 6N 29 / 34

30 Write down y 0 as a linear combination of eigenvectors, y 0 = c 1 v 1 + c 2 v 2 that is, solving Pc = y 0. So we have c 1 = 1/ 5 and c 2 = c 1. y 100 = c 1 λ v 1 + c 2 λ v 2 c 1 λ v 1 x λ = Xiaohui Xie (UCI) ICS 6N 30 / 34

31 Complex eigenvalues Find the eigenvalues of A = [ ] Solution: det(a λi ) = λ = 0 This means it doesn t have real solutions... So we have to use complex numbers Here λ 1 = 1 = i, and λ 2 = 1 = i where i = 1 and i 2 = 1 Xiaohui Xie (UCI) ICS 6N 31 / 34

32 Complex numbers z = a + bi where a and b are real a: represents the real part of z, denoted Re(z) b: represents the imaginary part of z, denoted Im(z) z 1 = z 2 if and only if Re(z 1 ) = Re(z 2 ) and Im(z 1 ) = Im(z 2 ) Given z 1 = a 1 + b 1 i, z 2 = a 2 + b 2 i, definite addition and multiplication by z 1 + z 2 = (a 1 + a 2 ) + (b 1 + b 2 )i z 1 z 2 = (a 1 a 2 b 1 b 2 ) + (a 1 b 2 + a 2 b 1 )i Xiaohui Xie (UCI) ICS 6N 32 / 34

33 Complex eigenvalues The characteristic equation det(a λi ) = 0 is exactly n roots if complex values are allowed. (λ λ 1 )... (λ λ n ) = 0 Xiaohui Xie (UCI) ICS 6N 33 / 34

34 Symmetric matrices A matrix is called symmetric if A = A T The eigenvalues of a symmetric matrix are all real. Xiaohui Xie (UCI) ICS 6N 34 / 34

ICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization

ICS 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 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 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

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

ICS 6N Computational Linear Algebra Vector Space

ICS 6N Computational Linear Algebra Vector Space ICS 6N Computational Linear Algebra Vector Space Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 24 Vector Space Definition: A vector space is a non empty set V of

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

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

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

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

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

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

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

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

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

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

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

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

ICS 6N Computational Linear Algebra Vector Equations

ICS 6N Computational Linear Algebra Vector Equations ICS 6N Computational Linear Algebra Vector Equations Xiaohui Xie University of California, Irvine xhx@uci.edu January 17, 2017 Xiaohui Xie (UCI) ICS 6N January 17, 2017 1 / 18 Vectors in R 2 An example

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

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 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

Eigenvalues and Eigenvectors

Eigenvalues 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 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

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

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

Numerical Linear Algebra Homework Assignment - Week 2

Numerical Linear Algebra Homework Assignment - Week 2 Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.

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

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

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

Final Review Written by Victoria Kala SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015

Final Review Written by Victoria Kala SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015 Final Review Written by Victoria Kala vtkala@mathucsbedu SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015 Summary This review contains notes on sections 44 47, 51 53, 61, 62, 65 For your final,

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

(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

1. General Vector Spaces

1. General Vector Spaces 1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

More information

a 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12

a 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12 24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2

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

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

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

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

Jordan Normal Form and Singular Decomposition

Jordan 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 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

A = 3 1. We conclude that the algebraic multiplicity of the eigenvalues are both one, that is,

A = 3 1. We conclude that the algebraic multiplicity of the eigenvalues are both one, that is, 65 Diagonalizable Matrices It is useful to introduce few more concepts, that are common in the literature Definition 65 The characteristic polynomial of an n n matrix A is the function p(λ) det(a λi) Example

More information

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 240 Spring, Chapter 1: Linear Equations and Matrices MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Eigenvalue Problems; Similarity Transformations Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 18 Eigenvalue

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

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

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

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

Math 315: Linear Algebra Solutions to Assignment 7

Math 315: Linear Algebra Solutions to Assignment 7 Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are

More information

Online Exercises for Linear Algebra XM511

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

More information

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

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

MATH 423 Linear Algebra II Lecture 20: Geometry of linear transformations. Eigenvalues and eigenvectors. Characteristic polynomial.

MATH 423 Linear Algebra II Lecture 20: Geometry of linear transformations. Eigenvalues and eigenvectors. Characteristic polynomial. MATH 423 Linear Algebra II Lecture 20: Geometry of linear transformations. Eigenvalues and eigenvectors. Characteristic polynomial. Geometric properties of determinants 2 2 determinants and plane geometry

More information

Math 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:

Math 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name: Math 34/84 - Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your

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

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

2. Every linear system with the same number of equations as unknowns has a unique solution.

2. Every linear system with the same number of equations as unknowns has a unique solution. 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

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

Practice Problems for the Final Exam

Practice Problems for the Final Exam Practice Problems for the Final Exam Linear Algebra. Matrix multiplication: (a) Problem 3 in Midterm One. (b) Problem 2 in Quiz. 2. Solve the linear system: (a) Problem 4 in Midterm One. (b) Problem in

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors November 3, 2016 1 Definition () The (complex) number λ is called an eigenvalue of the n n matrix A provided there exists a nonzero (complex) vector v such that Av = λv, in which case the vector v is called

More information

0.1 Eigenvalues and Eigenvectors

0.1 Eigenvalues and Eigenvectors 0.. EIGENVALUES AND EIGENVECTORS MATH 22AL Computer LAB for Linear Algebra Eigenvalues and Eigenvectors Dr. Daddel Please save your MATLAB Session (diary)as LAB9.text and submit. 0. 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

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

3 Matrix Algebra. 3.1 Operations on matrices

3 Matrix Algebra. 3.1 Operations on matrices 3 Matrix Algebra A matrix is a rectangular array of numbers; it is of size m n if it has m rows and n columns. A 1 n matrix is a row vector; an m 1 matrix is a column vector. For example: 1 5 3 5 3 5 8

More information

Section 8.2 : Homogeneous Linear Systems

Section 8.2 : Homogeneous Linear Systems Section 8.2 : Homogeneous Linear Systems Review: Eigenvalues and Eigenvectors Let A be an n n matrix with constant real components a ij. An eigenvector of A is a nonzero n 1 column vector v such that Av

More information

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

More information

Math 369 Exam #2 Practice Problem Solutions

Math 369 Exam #2 Practice Problem Solutions Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.

More information

Calculating determinants for larger matrices

Calculating determinants for larger matrices Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector

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

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) AMS526: Numerical Analysis (Numerical Linear Algebra for Computational and Data Sciences) Lecture 14: Eigenvalue Problems; Eigenvalue Revealing Factorizations Xiangmin Jiao Stony Brook University Xiangmin

More information

Math 121 Practice Final Solutions

Math 121 Practice Final Solutions Math Practice Final Solutions December 9, 04 Email me at odorney@college.harvard.edu with any typos.. True or False. (a) If B is a 6 6 matrix with characteristic polynomial λ (λ ) (λ + ), then rank(b)

More information

UNIT 6: The singular value decomposition.

UNIT 6: The singular value decomposition. UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 2011-2012 A square matrix is symmetric if A T

More information

MATH 221, Spring Homework 10 Solutions

MATH 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 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

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

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 1553-C MIDTERM EXAMINATION 3

MATH 1553-C MIDTERM EXAMINATION 3 MATH 553-C MIDTERM EXAMINATION 3 Name GT Email @gatech.edu Please read all instructions carefully before beginning. Please leave your GT ID card on your desk until your TA scans your exam. Each problem

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

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

Unit 5. Matrix diagonaliza1on

Unit 5. Matrix diagonaliza1on Unit 5. Matrix diagonaliza1on Linear Algebra and Op1miza1on Msc Bioinforma1cs for Health Sciences Eduardo Eyras Pompeu Fabra University 218-219 hlp://comprna.upf.edu/courses/master_mat/ We have seen before

More information

Jordan Canonical Form Homework Solutions

Jordan 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 information

CHAPTER 3. Matrix Eigenvalue Problems

CHAPTER 3. Matrix Eigenvalue Problems A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL

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

TBP MATH33A Review Sheet. November 24, 2018

TBP MATH33A Review Sheet. November 24, 2018 TBP MATH33A Review Sheet November 24, 2018 General Transformation Matrices: Function Scaling by k Orthogonal projection onto line L Implementation If we want to scale I 2 by k, we use the following: [

More information

Linear Algebra - Part II

Linear Algebra - Part II Linear Algebra - Part II Projection, Eigendecomposition, SVD (Adapted from Sargur Srihari s slides) Brief Review from Part 1 Symmetric Matrix: A = A T Orthogonal Matrix: A T A = AA T = I and A 1 = A T

More information

Math 21b Final Exam Thursday, May 15, 2003 Solutions

Math 21b Final Exam Thursday, May 15, 2003 Solutions Math 2b Final Exam Thursday, May 5, 2003 Solutions. (20 points) True or False. No justification is necessary, simply circle T or F for each statement. T F (a) If W is a subspace of R n and x is not in

More information

Mathematical Methods for Engineers 1 (AMS10/10A)

Mathematical Methods for Engineers 1 (AMS10/10A) Mathematical Methods for Engineers 1 (AMS10/10A) Quiz 5 - Friday May 27th (2016) 2:00-3:10 PM AMS 10 AMS 10A Name: Student ID: Multiple Choice Questions (3 points each; only one correct answer per question)

More information

Symmetric and anti symmetric matrices

Symmetric and anti symmetric matrices Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal

More information

Math Final December 2006 C. Robinson

Math Final December 2006 C. Robinson Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the

More information

Lecture Summaries for Linear Algebra M51A

Lecture Summaries for Linear Algebra M51A These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture

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

MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.

MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible. MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:

More information

235 Final exam review questions

235 Final exam review questions 5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an

More information

Homework sheet 4: EIGENVALUES AND EIGENVECTORS. DIAGONALIZATION (with solutions) Year ? Why or why not? 6 9

Homework 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 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

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

(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

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