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


 Alban Camron Dawson
 1 years ago
 Views:
Transcription
1 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 is much more straightforward Diagonal matrices are particularly nice For example, the product of two diagonal matrices can be computed by simply multiplying their corresponding diagonal entries: a b a b a 2 b 2 a 2 b 2 a n b n a n b n Computationally, diagonal matrices are the easiest to work with With this idea in mind, we introduce similarity: Definition An n n matrix A is similar to a matrix B if there is an invertible matrix P so that B P AP, and the function c P defined by is called a similarity transformation c P (A) P AP As an example, matrices are similar; with you should check that A P ( ) 4 ( 4 ) 3 and B and P ( 8 ) ( ), 3 4 ( 8 ) so that A and B are indeed similar B P AP ( ) ( ) ( 4 3 We are concerned with similarity not for its own sake as an interesting phenomenon, but because of quantities known as similarity invariants We can get a feel for what similarity invariants are by considering data about the matrices from the previous example: in particular, let s calculate the determinant, trace and eigenvalues of A and B: ),
2 property A B determinant 4 4 trace 5 5 eigenvalues, 4, 4 As you may have guessed from the table above, certain matrix properties, such as the determinant, trace, and eigenvalues, are shared among similar matrices; this is what we mean when we use the phrase similarity invariant In other words, the determinant of the matrix A is also the determinant of any matrix similar to A In fact, it is quite easy to check that the determinant is a similarity invariant; to do so, we recall two rules: det(ab) det(a) det(b), and det(p ) det P Any matrix similar to a given matrix A must have form P AP, with P an invertible matrix Let s calculate the determinant of this similar matrix, using the rules above: det(p AP ) (det P )(det A)(det P ) ( ) (det A)(det P ) det P det A We have just shown that similar matrices share determinants that is, det A det(p AP ) for any invertible matrix P In general, if matrices A and B are similar, so that B P AP, then they share: determinant trace eigenvalues rank nullity invertibility characteristic polynomial eigenspace corresponding to a particular (shared) eigenvalue Based on the example above, you may have already guessed the reason that we care about the idea of similarity: If A and B are similar matrices, and if A is diagonal, then it is much easier to calculate data such as determinant and eigenvalues about A than it is about B With this in mind, we introduce the idea of diagonalizability: Definition 2 An n n matrix A is diagonalizable if it is similar to a diagonal matrix That is, A is diagonalizable if there is an invertible matrix P so that P AP is diagonal The matrix B ( 8 )
3 is diagonalizable, since it is similar to the diagonal matrix ( ) A 4 Key Point It is important to note that not every matrix is diagonalizable Indeed, there are many matrices which are simply not similar to a diagonal matrix We will examine a few later in this section Criteria for Diagonalizability Given the information we have collected so far, it should be clear that, while diagonal matrices are arguably the easiest type of matrix to work with, diagonalizable matrices are almost as easy If I want to know the determinant, trace, etc of a matrix B that is similar to a diagonal matrix A, then I simply need to make the (easier) calculations for A This leads to a few interesting questions: how can we be certain that a given matrix is diagonalizable? And if we know that a matrix is diagonlizable, how do we find the diagonal matrix to which it is similar? The following theorem answers the first question: Theorem 52 If A is an n n matrix, then the following statements are equivalent: (a) A is diagonalizable (b) A has n linearly independent eigenvectors In other words, we can check that matrix A is diagonalizable by looking at its eigenvectors: if A has n linearly independent eigenvectors, then it is diagonalizable Example In Section 5, we saw that the matrix 2 A has repeated eigenvalue λ λ 3 2, with two corresponding linearly independent eigenvectors x and x 3 4 3
4 In addition, the eigenvalues λ 2 and λ 4 3 have eigenvectors x 2 4 and x 4, 3 respectively You should check that the four eigenvectors above are linearly independent inspecting the linear combination ax + bx 2 + cx 3 + dx 4 ; indeed, it is easy to see that the corresponding system a b 4 3 b + c 4a + d has only the trivial solution a b c d Since A is 4 4 and has 4 distinct eigenvectors, A is diagonalizable Example Determine if the matrix is diagonalizable A ( ) 4 2 We need to check the eigenvectors of A; if A is diagonalizable, then it has two linearly independent eigenvectors Accordingly, we begin by finding the eigenvalues of A, using the characteristic equation: ( ) λ 4 det(λi A) det λ 2 (λ 4)(λ 2) + λ 2 6λ λ 2 6λ + 9, 4
5 so that the characteristic equation for A is λ 2 6λ + 9 By factoring the equation, we see that its roots are λ 3, so that A has a single repeated eigenvalue Any eigenvector x corresponding to λ 3 must satisfy Ax 3x ( ) ( ) 4 x 2 x 2 ( ) 4x x 2 x + 2x 2 ( ) 3x 3x 2 ( ) 3x 3x 2 Thus we see that both of which amount to the single equation 4x x 2 3x x + 2x 2 3x 2, x x 2 or x x 2 Parameterizing x as x t, we see that any eigenvector of A must have form ( ) ( ) t t t Thus A has only one linearly independent eigenvector; since A is 2 2, it is not diagonalizable The following theorem on eigenvalues and their associated eigenvectors will give us a quick way to check some matrices for diagonalizability: Theorem 522 If λ, λ 2,, λ k are distinct eigenvalues of an n n matrix A, and x, x 2,, x k are eigenvectors corresponding to λ, λ 2,, λ k respectively, then the set is a linearly independent set {x, x 2,, x k } The theorem says that, for each distinct eigenvalue of a matrix A, we are guaranteed another linearly independent eigenvector For example, if a 4 4 matrix A has eigenvalues, 2,, and 5, then since it has four distinct eigenvalues, A automatically has four linearly independent eigenvectors Taken together with Theorem 52 on diagonlizability, we have the following corollary: Corollary Any n n matrix with distinct eigenvalues is diagonalizable 5
6 Key Point We must be extremely careful to note that we can only use the corollary to draw conclusions about an n n matrix if the matrix has n distinct eigenvalues If the matrix does not have n distinct eigenvalues, then it may or may not be diagonalizable In fact, we have seen two matrices 2 ( ) and with repeated eigenvalues: the first matrix has eigenvalues, 2, 2, and 3, and the second has eigenvalues 3 and 3 The first matrix is diagonalizable, while the second is not Finding the Similar Diagonal Matrix Earlier, we asked how we could go about finding the diagonal matrix to which a diagonalizable matrix A is similar If A is diagonalizable, with P AP the desired diagonal matrix, then we can rephrase the question above: How do we find P? The answer to this question turns out to be quite interesting: Theorem Let the n n matrix A be diagonalizable with n linearly independent eigenvectors x, x 2,, x n Set P x x 2 x n In other words, P is the matrix whose columns are the n linearly independent eigenvectors of A Then P AP is a diagonal matrix whose diagonal entries are the eigenvalues λ, λ 2,, λ n that correspond to the eigenvectors forming the successive columns of P Example Find the diagonal matrix to which is similar Earlier, we saw that matrix 2 A A
7 has linearly independent eigenvectors x corresponding to eigenvalues 4, x 2 4 3, x 3, and x 4 λ 2, λ 2, λ 3 2, and λ 4 3 respectively The matrix P from the theorem above is given by you should check that P 4 ; 3 4 P is its inverse According to the theorem, P AP is diagonal let s verify this: 2 P AP
8 Powers of a Diagonalizable Matrix Matrix multiplication with diagonal matrices is remarkably simple As a quick example, we can calculate the product below simply by multiplying corresponding diagonal entries: As you might have guessed, there is a simple formula for calculating powers of diagonal matrices: if d d 2 D, d n then D k exists if k is any integer and each d i is nonzero, or one or more of the d i is, and k is a positive integer If D k exists, then d k D k d k 2 d k n We can use these facts to our advantage if we know that matrix A is diagonalizable To see why, suppose we wished to calculate A k, where A is diagonalizable but not itself diagonal Since A is diagonalizable, there is a diagonal matrix D and invertible matrix P so that thus we rewrite A k as A k (P DP ) k A P DP ; (P DP )(P DP )(P DP ) (P DP )(P DP ) }{{} k copies P D(P P )D(P P )D(P P )D(P P )DP ) P DDD }{{ DD} P k copies P D k P 8
9 In other words, in order to calculate A k (which might be hard), we can first calculate D k (easy), then finish off with a similarity transformation: A k P D k P Key Point The idea behind the observations above are actually true in a more general context than the one presented above: if A and B (not necessarily diagonal) are similar by an invertible matrix P, then A k and B k are also similar by P Example Given find the eigenvalues of A 5 2 A 4 2, 4 3 In an earlier example, we saw that A is diagonalizable via P to the diagonal matrix 2 D 2 3 Now since A and D are similar, they share eigenvalues; A 5 and D 5 are also similar (via the same matrix P ), so that they share eigenvalues as well The eigenvalues of D 5 are easy to find; D 5 is diagonal, so its eigenvalues are its diagonal entries Let s make the calculation: 2 5 D 5 ( )
10 Thus D 5 has eigenvalues λ 32, λ 2, λ 3 32, and λ 4 243, which it shares with A 5 since they are similar Geometric and Algebraic Multiplicity Given an n n matrix A, there are two possibilities for the types of eigenvalues A could have: n distinct eigenvalues 2 some repeated eigenvalues In the first case, we automatically know that A is diagonalizable; however, in the second case, A may or may not be diagonalizable Indeed, we saw two different examples of repeated eigenvalues: 2 A has eigenvalues, 2, 2, and 3 and is diagonalizable, while ( ) 4 B 2 has eigenvalues 3 and 3 but is not diagonalizable There are apparently different types of repeated eigenvalues those, like the 2 from matrix A above, that do not affect diagonalizability, and those such as the 3 from matrix B which do We would like to have some terminology to help us differentiate between good eigenvalues and bad ones, and so we will momentarily introduce the ideas of algebraic and geometric multiplicity Before we do so, recall that eigenvalues are nothing but roots of the characteristic equation of the associated matrix For example, since matrix B above has eigenvalue 3 repeated twice, its characteristic equation must be (λ 3)(λ 3) Definition Let λ be an eigenvalue of the matrix A The number of times that the factor (λ λ ) appears as a factor of the characteristic polynomial det(λi A) of A is called the algebraic multiplicity of λ The dimension of the eigenspace associated with λ is called the geometric multiplicity of λ Let s think about the ideas of algebraic and geometric multiplicity in terms of the two examples above Starting with 2 A 4 2, 4 3
11 we can calculate that its characteristic equation is (λ 2)(λ 2)(λ + )(λ 3) Since λ 2 shows up twice as a factor, the eigenvalue 2 has algebraic multiplicity 2; the other two eigenvalues have algebraic multiplicity To get the geometric multiplicities of the eigenvalues, we need to check the eigenspaces corresponding to each distinct eigenvalue: earlier, we saw that A has linearly independent eigenvectors x 4, x 2 4 3, x 3, and x 4 ; x and x 3 correspond to the repeated eigenvalue λ 2, x 2 corresponds to the distinct eigenvalue λ 2 and x 4 corresponds to the distinct eigenvalue λ 4 3 Thus the repeated eigenvalue 2 corresponds to an eigenspace with a basis consisting of 2 vectors, whereas each of the eigenvalues and 3 have eigenspaces with a basis consisting of vector Thus the geometric multiplicity of the eigenvalue 2 is 2, and the eigenvalues and 3 both have geometric multiplicities We record all of the data in a table: Distinct Geometric Algebraic Eigenvalue Multiplicity Multiplicity Next, let s examine matrix B It is easy to calculate the characteristic equation ( ) 4 2 (λ 3)(λ 3) ; since the factor (λ 3) shows up twice in the equation, the repeated eigenvalue λ 3 has algebraic multiplicity 2 Earlier, we saw that the eigenspace corresponding to eigenvalue λ 3 is generated by the single vector ( ) x ; thus λ 3 has geometric multiplicity Let s compare all of the data that we have generated:
12 Distinct Geometric Algebraic Distinct Geometric Algebraic Eigenvalue of A Multiplicity Multiplicity Eigenvalue of B Multiplicity Multiplicity Notice in the example above that the algebraic and geometric multiplicities of each distinct eigenvalue of the diagonalizable matrix A matched up, whereas the geometric multiplicity of the only distinct eigenvalue of the nondiagonalizable matrix B was deficient compared with its algebraic multiplicity This idea is actually a general rule, made precise in the following theorem: Theorem 524 Let A be an n n matrix (a) The geometric multiplicity of a distinct eigenvalue of A is less than or equal to its algebraic multiplicity (b) A is diagonalizable if and only if the algebraic and geometric multiplicities of each of its distinct eigenvalues are equal 2
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 informationDiagonalization. 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 informationRemark 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 informationMath 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 informationDIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix
DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that
More informationTherefore, 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 informationRemark 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 informationQuestion: 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 AcI?
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 AcI? Property The eigenvalues
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture 9: Diagonalization Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./9 Section. Diagonalization The goal here is to develop a useful
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationDefinition (T invariant subspace) Example. Example
Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin
More information(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 informationAMS10 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 information2. 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 informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.15.4, 6.16.2 and 7.17.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationSolutions to Final Exam
Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5vector. Assume that the nullity of A is. (a) What is the rank of A? 3 (b) Are the rows of A linearly independent? (c) Are the columns
More informationc 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 informationLINEAR ALGEBRA 1, 2012I 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 informationChapter 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 information1. In this problem, if the statement is always true, circle T; otherwise, circle F.
Math 1553, Extra Practice for Midterm 3 (sections 4565) 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 informationExamples 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 informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors week 2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n
More informationCalculating 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 informationUNDERSTANDING THE DIAGONALIZATION PROBLEM. Roy Skjelnes. 1. Linear Maps 1.1. Linear maps. A map T : R n R m is a linear map if
UNDERSTANDING THE DIAGONALIZATION PROBLEM Roy Skjelnes Abstract These notes are additional material to the course B107, given fall 200 The style may appear a bit coarse and consequently the student is
More informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationA = 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 informationMAT1302F Mathematical Methods II Lecture 19
MAT302F Mathematical Methods II Lecture 9 Aaron Christie 2 April 205 Eigenvectors, Eigenvalues, and Diagonalization Now that the basic theory of eigenvalues and eigenvectors is in place most importantly
More information2 b 3 b 4. c c 2 c 3 c 4
OHSx XM511 Linear Algebra: Multiple Choice Questions for Chapter 4 a a 2 a 3 a 4 b b 1. What is the determinant of 2 b 3 b 4 c c 2 c 3 c 4? d d 2 d 3 d 4 (a) abcd (b) abcd(a b)(b c)(c d)(d a) (c) abcd(a
More informationEigenvalues, Eigenvectors, and Diagonalization
Math 240 TA: Shuyi Weng Winter 207 February 23, 207 Eigenvalues, Eigenvectors, and Diagonalization The concepts of eigenvalues, eigenvectors, and diagonalization are best studied with examples. We will
More informationEigenspaces and Diagonalizable Transformations
Chapter 2 Eigenspaces and Diagonalizable Transformations As we explored how heat states evolve under the action of a diffusion transformation E, we found that some heat states will only change in amplitude.
More informationUnit 5: Matrix diagonalization
Unit 5: Matrix diagonalization Juan Luis Melero and Eduardo Eyras October 2018 1 Contents 1 Matrix diagonalization 3 1.1 Definitions............................. 3 1.1.1 Similar matrix.......................
More informationDiagonalization of Matrix
of Matrix King Saud University August 29, 2018 of Matrix Table of contents 1 2 of Matrix Definition If A M n (R) and λ R. We say that λ is an eigenvalue of the matrix A if there is X R n \ {0} such that
More informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationEigenvalues and Eigenvectors 7.2 Diagonalization
Eigenvalues and Eigenvectors 7.2 Diagonalization November 8 Goals Suppose A is square matrix of order n. Provide necessary and sufficient condition when there is an invertible matrix P such that P 1 AP
More informationMath 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 informationDeterminants and Scalar Multiplication
Invertibility and Properties of Determinants In a previous section, we saw that the trace function, which calculates the sum of the diagonal entries of a square matrix, interacts nicely with the operations
More information235 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 informationReview 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 informationCity 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 informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationDiagonalisierung. Eigenwerte, Eigenvektoren, Mathematische Methoden der Physik I. Vorlesungsnotizen zu
Eigenwerte, Eigenvektoren, Diagonalisierung Vorlesungsnotizen zu Mathematische Methoden der Physik I J. Mark Heinzle Gravitational Physics, Faculty of Physics University of Vienna Version /6/29 2 version
More informationLecture 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 informationLinear 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 informationMAC Module 12 Eigenvalues and Eigenvectors. Learning Objectives. Upon completing this module, you should be able to:
MAC Module Eigenvalues and Eigenvectors Learning Objectives Upon completing this module, you should be able to: Solve the eigenvalue problem by finding the eigenvalues and the corresponding eigenvectors
More informationMAC Module 12 Eigenvalues and Eigenvectors
MAC 23 Module 2 Eigenvalues and Eigenvectors Learning Objectives Upon completing this module, you should be able to:. Solve the eigenvalue problem by finding the eigenvalues and the corresponding eigenvectors
More informationDimension. Eigenvalue and eigenvector
Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, ranknullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,
More information= main diagonal, in the order in which their corresponding eigenvectors appear as columns of E.
3.3 Diagonalization Let A = 4. Then and are eigenvectors of A, with corresponding eigenvalues 2 and 6 respectively (check). This means 4 = 2, 4 = 6. 2 2 2 2 Thus 4 = 2 2 6 2 = 2 6 4 2 We have 4 = 2 0 0
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationMath 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 informationGeneralized Eigenvectors and Jordan Form
Generalized Eigenvectors and Jordan Form We have seen that an n n matrix A is diagonalizable precisely when the dimensions of its eigenspaces sum to n. So if A is not diagonalizable, there is at least
More informationMath 110 Linear Algebra Midterm 2 Review October 28, 2017
Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections
More informationMath 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 informationEigenvalues and Eigenvectors: An Introduction
Eigenvalues and Eigenvectors: An Introduction The eigenvalue problem is a problem of considerable theoretical interest and wideranging application. For example, this problem is crucial in solving systems
More informationHomework sheet 4: EIGENVALUES AND EIGENVECTORS. DIAGONALIZATION (with solutions) Year ? Why or why not? 6 9
Bachelor in Statistics and Business Universidad Carlos III de Madrid Mathematical Methods II María Barbero Liñán Homework sheet 4: EIGENVALUES AND EIGENVECTORS DIAGONALIZATION (with solutions) Year  Is
More informationDeterminants and Scalar Multiplication
Properties of Determinants In the last section, we saw how determinants interact with the elementary row operations. There are other operations on matrices, though, such as scalar multiplication, matrix
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationMATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization.
MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization. Eigenvalues and eigenvectors of an operator Definition. Let V be a vector space and L : V V be a linear operator. A number λ
More informationUnit 5. Matrix diagonaliza1on
Unit 5. Matrix diagonaliza1on Linear Algebra and Op1miza1on Msc Bioinforma1cs for Health Sciences Eduardo Eyras Pompeu Fabra University 218219 hlp://comprna.upf.edu/courses/master_mat/ We have seen before
More informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationMATH 1553 PRACTICE MIDTERM 3 (VERSION B)
MATH 1553 PRACTICE MIDTERM 3 (VERSION B) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.
More informationSolutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015
Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See
More informationEigenvalues 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(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 informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More informationDM554 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 informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationREVIEW FOR EXAM III SIMILARITY AND DIAGONALIZATION
REVIEW FOR EXAM III The exam covers sections 4.4, the portions of 4. on systems of differential equations and on Markov chains, and..4. SIMILARITY AND DIAGONALIZATION. Two matrices A and B are similar
More informationFinal 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(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 informationMATH 1553C MIDTERM EXAMINATION 3
MATH 553C 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 informationEigenvalues and Eigenvectors
Sec. 6.1 Eigenvalues and Eigenvectors Linear transformations L : V V that go from a vector space to itself are often called linear operators. Many linear operators can be understood geometrically by identifying
More informationEigenvalues 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 informationEigenvalues 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 informationEigenvalues 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 informationMATH 235. Final ANSWERS May 5, 2015
MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your
More informationAnnouncements Wednesday, November 7
Announcements Wednesday, November 7 The third midterm is on Friday, November 6 That is one week from this Friday The exam covers 45, 5, 52 53, 6, 62, 64, 65 (through today s material) WeBWorK 6, 62 are
More informationMATH 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 informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKERKAHN 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 informationftuiowamath2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST
me me ftuiowamath2550 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 information5.3.5 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 informationLecture 15, 16: Diagonalization
Lecture 15, 16: Diagonalization Motivation: Eigenvalues and Eigenvectors are easy to compute for diagonal matrices. Hence, we would like (if possible) to convert matrix A into a diagonal matrix. Suppose
More informationFinal A. Problem Points Score Total 100. Math115A Nadja Hempel 03/23/2017
Final A Math115A Nadja Hempel 03/23/2017 nadja@math.ucla.edu Name: UID: Problem Points Score 1 10 2 20 3 5 4 5 5 9 6 5 7 7 8 13 9 16 10 10 Total 100 1 2 Exercise 1. (10pt) Let T : V V be a linear transformation.
More informationProblem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show
MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology  Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,
More information1. 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 informationMath 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 information8. Diagonalization.
8. Diagonalization 8.1. Matrix Representations of Linear Transformations Matrix of A Linear Operator with Respect to A Basis We know that every linear transformation T: R n R m has an associated standard
More informationWarmup. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions
Warmup 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 informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #2 Solutions
YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 3. M Test # Solutions. (8 pts) For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For this
More informationMath 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 informationMath 2030, Matrix Theory and Linear Algebra I, Fall 2011 Final Exam, December 13, 2011 FIRST NAME: LAST NAME: STUDENT ID:
Math 2030, Matrix Theory and Linear Algebra I, Fall 20 Final Exam, December 3, 20 FIRST NAME: LAST NAME: STUDENT ID: SIGNATURE: Part I: True or false questions Decide whether each statement is true or
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall 2015 1 / 14 Introduction We define eigenvalues and eigenvectors. We discuss how to
More information4. 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 informationAnnouncements 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 informationLinear 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 informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationEIGENVALUES AND EIGENVECTORS
EIGENVALUES AND EIGENVECTORS Diagonalizable linear transformations and matrices Recall, a matrix, D, is diagonal if it is square and the only nonzero entries are on the diagonal This is equivalent to
More informationMATH 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