CAAM 335 Matrix Analysis

Size: px
Start display at page:

Download "CAAM 335 Matrix Analysis"

Transcription

1 CAAM 335 Matrix Analysis Solutions to Homework 8 Problem (5+5+5=5 points The partial fraction expansion of the resolvent for the matrix B = is given by (si B = s } {{ } =P + s + } {{ } =P + (s (5 points Write the P j s into a vector outer-product form: P j = v j w T j for j =,,3. (5 points Give 3 eigen-pairs, (λ j,x j, of B with right eigen-vectors x j C 3. (5 points Give 3 eigen-pairs, (λ j,y j, of B with left eigen-vectors y j C 3. The eigenvalues are, and. P = = P = P 3 = = = (, v = (, v = (, v 3 = The v j s are right eigenvectors (so are any nonzero multiples of them. λ =, x = ; λ =, x =. } {{ } =P 3, w =, w =, w 3 = ; λ 3 =, x 3 =

2 The w j s are left eigenvectors (so are any nonzero multiples of them. λ =, y = Problem (5+=5 points The solution of the dynamical system is given by x(t = exp(btx. ; λ =, y = ; λ 3 =, y 3 = x (t = Bx(t, t >, x( = x ( (a Let B R 3 3 be a matrix with three distinct eigenvalues λ,λ,λ 3. The partial fraction expansion of the resolvent is Since B = λ P + λ P + λ 3 P 3, we have R(s = (si B = s λ P + s λ P + s λ 3 P 3. tb = tλ P +tλ P +tλ 3 P 3. Apply the matrix exponential formula to exp(tb and express x(t = exp(btx in terms of t, x, P,P,P 3 and the eigenvalues. (b Let B be the matrix in problem and x = (,, T. Determine lim t x(t. From the notes, exp(b =e λ P + e λ P + e λ 3 P 3. thus exp(btx = (e tλ P + e tλ P + e tλ 3 P 3 x. exp(btx = e t = + e t + e t As t, the second term goes to zero and thus lim x(t = t (,,T. + e t

3 3 Problem 3 (+5+5= points a Compute by hand (show your steps the eigenvalues and eigenvectors of the matrix 4 A = 4. b What are the algebraic and geometric multiplicities of the eigenvalue(s? c Give Jordan blocks of this matrix. To determine the eigenvalues we compute the characteristic polynomial of A, λ + 4 p(λ = det(λi A = det 4 λ. λ Computing the determinant with respect to the last row or column gives, Thus A has one eigenvalue, λ =. p(λ = (λ 4λ + 4(λ = (λ 3. To determine the geometric multiplicity we determine the dimension of the nullspace of (A I. N (A I = N 4 = t + s : t,s R Thus λ = has algebraic multiplicity of 3 and a geometric multiplicity of. Based on the geometric multiplicity as computed above, the Jordan blocks of the matrix are as follows ( J = ( J = Problem Set 6. from the MIT Notes (p. 93 Online Problems, 3, 4, 8, 6, 7, 9, 9. 5 points each.. Find the eigenvalues and the eigenvectors of these two matrices: [ ] [ ] 4 4 A = and A + I = 3 4 A + I has the same eigenvectors as A. Its eigenvalues are increased by. ( λ 4 p A (λ = det λ 3 = λ 4λ 5 = (λ 5(λ + λ =,5

4 4 ( 4 (A + I = 4 x ( 4 4 (A 5I = x For the eigenvectors we see that ( λ 4 p A+I (λ = det λ 4 = λ 6λ = λ(λ 6 λ =,6 ((A + I I = (A + I ((A + I 6I = (A 5I, which were handled above. Thus A and A + I have the same eigenvectors. 3. Compute the eigenvalues and the eigenvectors A and A. Check the trace: [ ] [ ] A = and A = A has the same eigenvectors as A. When A has eigenvalues λ and λ, its inverse has eigenvalues, λ and λ. ( λ p A (λ = det λ = λ λ = (λ + (λ λ =, ( (A + I = x ( (A I = x The trace of A is + = + = λ + λ. ( λ + p A (λ = det λ For the eigenvectors we see that = λ + λ = (λ + (λ λ =, ( (A + I = x ( {}} { (A + I = ( x. The trace of A is + = + = λ + λ. Thus A and A have the same eigenvectors and the eigenvalues of A are the reciprocals of the eigenvalues of A.

5 5 4. Compute the eigenvalues and the eigenvectors A and A. Check the trace: [ ] [ ] 3 A = and A 7 3 = 6 A has the same eigenvectors as A. When A has eigenvalues λ and λ, A has eigenvalues λ and λ. In this example, why is λ + λ = 3? ( λ + 3 p A (λ = det = λ + λ 6 = (λ + 3(λ λ = 3, λ ( 3 (A + 3I = 3 x ( {}} { 3 ( 3 3 (A I = x ( λ 7 3 p A (λ = det λ 6 For the eigenvectors we see that = λ 3λ + 36 = (λ 9(λ 4 λ = 9,4 ( (A 3 9I = 3 x 3 ( (A 3 3 4I = x. Thus A and A have the same eigenvectors and the eigenvalues of A are the eigenvalues of A squared. Since the trace of the matrix is the sum of the diagonal entries as well as the sum of the eigenvalues, the trace of A is = λ + λ. 8. (a If you know that x is an eigenvector, the way to find λ is to compute Ax, the ratio between each entry of Ax and the corresponding entry of x should be constant and equal to λ. One could also compute x Ax/(x x, note x by definition. (b If you know that λ is an eigenvalue, the way to find x is to determine the null space of A λi. Any nonzero vector in the null-space of A λi is a suitable eigenvector. 6. The determinant of A equals the product λ λ λ n. Start with the polynomial det(a λi separated into is n factors (always possible. Then set λ = : det(a λi = (λ λ(λ λ...(λ n λ so deta = λ λ λ n Check this rule in Example where the Markov matrix has λ = and. Example [ ] λ = λ = det(a = (.8(.7 (.3(. = = λ λ

6 6 7. The sum of the diagonal entries (the trace equals the sum of the eigenvalues [ ] a b A = has det(a λi = λ (a + dλ + ad bc =. c d The quadratic formula gives the eigenvalues λ = (a + d + (a + d 4(ad bc/ and λ = (a + d (a + d 4(ad bc/. Their sum is a + d. If A has λ = 3 and λ = 4 then det(a λi = (λ 3(λ 4 = λ 7λ A 3 by 3 B is known to have eigenvalues,,. This information is enough to find three of these (give the answers where possible: the rank of B is [using, say, the expansion B = x y + x 3y 3 ] the determinant of B T B is [det(b T B = det(b = ( = ] the eigenvalues of B T B cannot be determined Take for example B = α B T B = α + for which B has the prescribed eigenvalues and B T B has eigenvalues,α + and with α being arbitrary. the eigenvalues of (B + I are,/ and /5 [by the formula /(λ + ]. 9. (Review Find the eigenvalues of A, B, and C: 3 A = 4 5, B = 6 3, C = Since A is upper triangular the eigenvalues are simply, 4, and 6. The characteristic polynomial of B is, det(λi B = λ(λ λ ( ( 3(λ = (λ (λ 3. hence the eigenvalues of B are, 3, and 3. 3 Let e T = [,, ]. Clearly C = ee T is a rank- and symmetric, and Ce = e(e T e = 6e. Hence the eigenvalues of C are 6 and where has a multiplicity of.

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

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

Conceptual Questions for Review

Conceptual Questions for Review Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.

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

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

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

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

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

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

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

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

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

EE263: Introduction to Linear Dynamical Systems Review Session 5

EE263: 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 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

HW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name.

HW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name. HW2 - Due 0/30 Each answer must be mathematically justified. Don t forget your name. Problem. Use the row reduction algorithm to find the inverse of the matrix 0 0, 2 3 5 if it exists. Double check your

More information

Math 18.6, Spring 213 Problem Set #6 April 5, 213 Problem 1 ( 5.2, 4). Identify all the nonzero terms in the big formula for the determinants of the following matrices: 1 1 1 2 A = 1 1 1 1 1 1, B = 3 4

More information

Systems of Algebraic Equations and Systems of Differential Equations

Systems of Algebraic Equations and Systems of Differential Equations Systems of Algebraic Equations and Systems of Differential Equations Topics: 2 by 2 systems of linear equations Matrix expression; Ax = b Solving 2 by 2 homogeneous systems Functions defined on matrices

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

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 review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

Eigenspaces. (c) Find the algebraic multiplicity and the geometric multiplicity for the eigenvaules of A.

Eigenspaces. (c) Find the algebraic multiplicity and the geometric multiplicity for the eigenvaules of A. Eigenspaces 1. (a) Find all eigenvalues and eigenvectors of A = (b) Find the corresponding eigenspaces. [ ] 1 1 1 Definition. If A is an n n matrix and λ is a scalar, the λ-eigenspace of A (usually denoted

More information

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax = . (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

More information

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

M 340L CS Homework Set 12 Solutions. Note: Scale all eigenvectors so the largest component is +1.

M 340L CS Homework Set 12 Solutions. Note: Scale all eigenvectors so the largest component is +1. M 34L CS Homework Set 2 Solutions Note: Scale all eigenvectors so the largest component is +.. For each of these matrices, find the characteristic polynomial p( ) det( A I). factor it to get the eigenvalues:,

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

(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

MATH 1553, C. JANKOWSKI MIDTERM 3

MATH 1553, C. JANKOWSKI MIDTERM 3 MATH 1553, C JANKOWSKI MIDTERM 3 Name GT Email @gatechedu Write your section number (E6-E9) here: Please read all instructions carefully before beginning Please leave your GT ID card on your desk until

More information

MIT Final Exam Solutions, Spring 2017

MIT Final Exam Solutions, Spring 2017 MIT 8.6 Final Exam Solutions, Spring 7 Problem : For some real matrix A, the following vectors form a basis for its column space and null space: C(A) = span,, N(A) = span,,. (a) What is the size m n of

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

Math 240 Calculus III

Math 240 Calculus III The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A

More information

Math Linear Algebra Final Exam Review Sheet

Math Linear Algebra Final Exam Review Sheet Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

More information

Determinants by Cofactor Expansion (III)

Determinants by Cofactor Expansion (III) Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...

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

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

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

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

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

1. Let A = (a) 2 (b) 3 (c) 0 (d) 4 (e) 1

1. Let A = (a) 2 (b) 3 (c) 0 (d) 4 (e) 1 . Let A =. The rank of A is (a) (b) (c) (d) (e). Let P = {a +a t+a t } where {a,a,a } range over all real numbers, and let T : P P be a linear transformation dedifined by T (a + a t + a t )=a +9a t If

More information

Math 304 Fall 2018 Exam 3 Solutions 1. (18 Points, 3 Pts each part) Let A, B, C, D be square matrices of the same size such that

Math 304 Fall 2018 Exam 3 Solutions 1. (18 Points, 3 Pts each part) Let A, B, C, D be square matrices of the same size such that Math 304 Fall 2018 Exam 3 Solutions 1. (18 Points, 3 Pts each part) Let A, B, C, D be square matrices of the same size such that det(a) = 2, det(b) = 2, det(c) = 1, det(d) = 4. 2 (a) Compute det(ad)+det((b

More information

Solutions Problem Set 8 Math 240, Fall

Solutions Problem Set 8 Math 240, Fall Solutions Problem Set 8 Math 240, Fall 2012 5.6 T/F.2. True. If A is upper or lower diagonal, to make det(a λi) 0, we need product of the main diagonal elements of A λi to be 0, which means λ is one of

More information

EE5120 Linear Algebra: Tutorial 6, July-Dec Covers sec 4.2, 5.1, 5.2 of GS

EE5120 Linear Algebra: Tutorial 6, July-Dec Covers sec 4.2, 5.1, 5.2 of GS EE0 Linear Algebra: Tutorial 6, July-Dec 07-8 Covers sec 4.,.,. of GS. State True or False with proper explanation: (a) All vectors are eigenvectors of the Identity matrix. (b) Any matrix can be diagonalized.

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

Linear Algebra Primer

Linear Algebra Primer Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................

More information

Linear algebra II Tutorial solutions #1 A = x 1

Linear algebra II Tutorial solutions #1 A = x 1 Linear algebra II Tutorial solutions #. Find the eigenvalues and the eigenvectors of the matrix [ ] 5 2 A =. 4 3 Since tra = 8 and deta = 5 8 = 7, the characteristic polynomial is f(λ) = λ 2 (tra)λ+deta

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

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

Introduction to Matrix Algebra

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

More information

Matrices and Linear Algebra

Matrices and Linear Algebra Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2

More information

6 EIGENVALUES AND EIGENVECTORS

6 EIGENVALUES AND EIGENVECTORS 6 EIGENVALUES AND EIGENVECTORS INTRODUCTION TO EIGENVALUES 61 Linear equations Ax = b come from steady state problems Eigenvalues have their greatest importance in dynamic problems The solution of du/dt

More information

Linear Algebra Exercises

Linear Algebra Exercises 9. 8.03 Linear Algebra Exercises 9A. Matrix Multiplication, Rank, Echelon Form 9A-. Which of the following matrices is in row-echelon form? 2 0 0 5 0 (i) (ii) (iii) (iv) 0 0 0 (v) [ 0 ] 0 0 0 0 0 0 0 9A-2.

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

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

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

PROBLEM SET. Problems on Eigenvalues and Diagonalization. Math 3351, Fall Oct. 20, 2010 ANSWERS

PROBLEM SET. Problems on Eigenvalues and Diagonalization. Math 3351, Fall Oct. 20, 2010 ANSWERS PROBLEM SET Problems on Eigenvalues and Diagonalization Math 335, Fall 2 Oct. 2, 2 ANSWERS i Problem. In each part, find the characteristic polynomial of the matrix and the eigenvalues of the matrix by

More information

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column

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

Chapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors

Chapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there

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

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

(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

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

Eigenvalues and Eigenvectors: An Introduction

Eigenvalues and Eigenvectors: An Introduction Eigenvalues and Eigenvectors: An Introduction The eigenvalue problem is a problem of considerable theoretical interest and wide-ranging application. For example, this problem is crucial in solving systems

More information

Linear algebra I Homework #1 due Thursday, Oct Show that the diagonals of a square are orthogonal to one another.

Linear algebra I Homework #1 due Thursday, Oct Show that the diagonals of a square are orthogonal to one another. Homework # due Thursday, Oct. 0. Show that the diagonals of a square are orthogonal to one another. Hint: Place the vertices of the square along the axes and then introduce coordinates. 2. Find the equation

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors CHAPTER Eigenvalues and Eigenvectors CHAPTER CONTENTS. Eigenvalues and Eigenvectors 9. Diagonalization. Complex Vector Spaces.4 Differential Equations 6. Dynamical Systems and Markov Chains INTRODUCTION

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

Topic 1: Matrix diagonalization

Topic 1: Matrix diagonalization Topic : Matrix diagonalization Review of Matrices and Determinants Definition A matrix is a rectangular array of real numbers a a a m a A = a a m a n a n a nm The matrix is said to be of order n m if it

More information

Linear Algebra 2 More on determinants and Evalues Exercises and Thanksgiving Activities

Linear Algebra 2 More on determinants and Evalues Exercises and Thanksgiving Activities Linear Algebra 2 More on determinants and Evalues Exercises and Thanksgiving Activities 2. Determinant of a linear transformation, change of basis. In the solution set of Homework 1, New Series, I included

More information

What s Eigenanalysis? Matrix eigenanalysis is a computational theory for the matrix equation

What s Eigenanalysis? Matrix eigenanalysis is a computational theory for the matrix equation Eigenanalysis What s Eigenanalysis? Fourier s Eigenanalysis Model is a Replacement Process Powers and Fourier s Model Differential Equations and Fourier s Model Fourier s Model Illustrated What is Eigenanalysis?

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

Background Mathematics (2/2) 1. David Barber

Background Mathematics (2/2) 1. David Barber Background Mathematics (2/2) 1 David Barber University College London Modified by Samson Cheung (sccheung@ieee.org) 1 These slides accompany the book Bayesian Reasoning and Machine Learning. The book and

More information

After we have found an eigenvalue λ of an n n matrix A, we have to find the vectors v in R n such that

After we have found an eigenvalue λ of an n n matrix A, we have to find the vectors v in R n such that 7.3 FINDING THE EIGENVECTORS OF A MATRIX After we have found an eigenvalue λ of an n n matrix A, we have to find the vectors v in R n such that A v = λ v or (λi n A) v = 0 In other words, we have to find

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

18.06SC Final Exam Solutions

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

More information

Math 314H Solutions to Homework # 3

Math 314H Solutions to Homework # 3 Math 34H Solutions to Homework # 3 Complete the exercises from the second maple assignment which can be downloaded from my linear algebra course web page Attach printouts of your work on this problem to

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

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 1 Exam 1 Solutions 6/12/3

Linear Algebra 1 Exam 1 Solutions 6/12/3 Linear Algebra 1 Exam 1 Solutions 6/12/3 Question 1 Consider the linear system in the variables (x, y, z, t, u), given by the following matrix, in echelon form: 1 2 1 3 1 2 0 1 1 3 1 4 0 0 0 1 2 3 Reduce

More information

EIGENVALUES AND EIGENVECTORS

EIGENVALUES AND EIGENVECTORS EIGENVALUES AND EIGENVECTORS Diagonalizable linear transformations and matrices Recall, a matrix, D, is diagonal if it is square and the only non-zero entries are on the diagonal This is equivalent 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

ELE/MCE 503 Linear Algebra Facts Fall 2018

ELE/MCE 503 Linear Algebra Facts Fall 2018 ELE/MCE 503 Linear Algebra Facts Fall 2018 Fact N.1 A set of vectors is linearly independent if and only if none of the vectors in the set can be written as a linear combination of the others. Fact N.2

More information

Math Matrix Algebra

Math Matrix Algebra Math 44 - Matrix Algebra Review notes - (Alberto Bressan, Spring 7) sec: Orthogonal diagonalization of symmetric matrices When we seek to diagonalize a general n n matrix A, two difficulties may arise:

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=

More information

Eigenvalues and Eigenvectors. Review: Invertibility. Eigenvalues and Eigenvectors. The Finite Dimensional Case. January 18, 2018

Eigenvalues and Eigenvectors. Review: Invertibility. Eigenvalues and Eigenvectors. The Finite Dimensional Case. January 18, 2018 January 18, 2018 Contents 1 2 3 4 Review 1 We looked at general determinant functions proved that they are all multiples of a special one, called det f (A) = f (I n ) det A. Review 1 We looked at general

More information

1. Select the unique answer (choice) for each problem. Write only the answer.

1. Select the unique answer (choice) for each problem. Write only the answer. MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +

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

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

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

More information

Matrix Solutions to Linear Systems of ODEs

Matrix Solutions to Linear Systems of ODEs Matrix Solutions to Linear Systems of ODEs James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 3, 216 Outline 1 Symmetric Systems of

More information

1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?

1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true? . Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in

More information

Systems of Second Order Differential Equations Cayley-Hamilton-Ziebur

Systems of Second Order Differential Equations Cayley-Hamilton-Ziebur Systems of Second Order Differential Equations Cayley-Hamilton-Ziebur Characteristic Equation Cayley-Hamilton Cayley-Hamilton Theorem An Example Euler s Substitution for u = A u The Cayley-Hamilton-Ziebur

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

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

Math 205, Summer I, Week 4b: Continued. Chapter 5, Section 8

Math 205, Summer I, Week 4b: Continued. Chapter 5, Section 8 Math 205, Summer I, 2016 Week 4b: Continued Chapter 5, Section 8 2 5.8 Diagonalization [reprint, week04: Eigenvalues and Eigenvectors] + diagonaliization 1. 5.8 Eigenspaces, Diagonalization A vector v

More information

Linear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway

Linear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway Linear Algebra: Lecture Notes Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway November 6, 23 Contents Systems of Linear Equations 2 Introduction 2 2 Elementary Row

More information

2 Eigenvectors and Eigenvalues in abstract spaces.

2 Eigenvectors and Eigenvalues in abstract spaces. MA322 Sathaye Notes on Eigenvalues Spring 27 Introduction In these notes, we start with the definition of eigenvectors in abstract vector spaces and follow with the more common definition of eigenvectors

More information

MAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:

MAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to: MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor

More information

Math 215 HW #9 Solutions

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

More information

Examples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions.

Examples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions. The exam will cover Sections 6.-6.2 and 7.-7.4: True/False 30% Definitions 0% Computational 60% Skip Minors and Laplace Expansion in Section 6.2 and p. 304 (trajectories and phase portraits) in Section

More information

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left

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

MAT2342 : Introduction to Linear Algebra Mike Newman, 5 October assignment 1

MAT2342 : Introduction to Linear Algebra Mike Newman, 5 October assignment 1 [/8 MAT4 : Introduction to Linear Algebra Mike Newman, 5 October 07 assignment You must show your work. You may use software or solvers of some sort to check calculation of eigenvalues, but you should

More information