The Exponential of a Matrix
|
|
- Silvester Young
- 5 years ago
- Views:
Transcription
1 The Exponential of a Matrix 5-8- The solution to the exponential growth equation dx dt kx is given by x c e kt It is natural to ask whether you can solve a constant coefficient linear system x A x in a similar way If a solution to the system is to have the same form as the growth equation solution, it should look like x e At x The first thing I need to do is to make sense of the matrix exponential e At The Taylor series for e z is e z z n It converges absolutely for all z It A is an n n matrix with real entries, define e At n t n A n n The powers A n make sense, since A is a square matrix It is possible to show that this series converges for all t and every matrix A Differentiating the series term-by-term, d dt eat n n tn A n n t n A n (n )! A n t n A n (n )! A m t m A m m! Ae At This shows that e At solves the differential equation x A x The initial condition vector x() x yields the particular solution x e At x This works, because e A I (by setting t in the power series) Another familiar property of ordinary exponentials holds for the matrix exponential: If A and B commute (that is, AB BA), then e A e B e A+B You can prove this by multiplying the power series for the exponentials on the left (e A is just e At with t ) ExampleCompute e At if A 3 Compute the successive powers of A: A 3, A 4,,A n n 9 3 n
2 e At n t n n 3 n n (t) n n (3t) n e t e 3t You can compute the exponential of an arbitrary diagonal matrix in the same way: λ λ A λ n, eat e λ t e λ t e λ nt Example Compute e At if A Compute the successive powers of A: A, A 4, A 3 6,,A n n Hence, e A n t n n Here s where the last equality came from: n t n n n nt n t n e t te t e t n t n et, n nt n t n t n (n )! t m t m m! tet Example Compute e At, if A 3 4 If you compute powers of A as in the last two examples, there is no evident pattern it would be difficult to compute the exponential using the power series Instead, set up the system whose coefficient matrix is A: x 3x y, y x 4y The solution is x c e t +c e t, y 5 c e t + c e t
3 Next, note that if B is a matrix, B first column of B and B In particular, this is true for e At Now is the solution satisfying x() x, but x Set x() (,) to get the first column of e At : x e At x c e t +c e t 5 c e t + c e t second column of B c +c 5 c + c Hence, c 5 3, c 3 So x y 5 Set x() (,) to get the second column of e At : c 3, c 3 Hence, 3 et 3 e t 3 et 3 e t c +c 5 c + c x y 3 et + 3 e t 3 et e t 5 e At 3 et 3 e t 3 et + 3 e t 3 et 3 e t 3 et e t I found e At, but I had to solve a system of differential equations in order to do it In some cases, it s possible to use linear algebra to compute the exponential of a matrix An n n matrix A is diagonalizable if it has n independent eigenvectors (This is true, for example, if A has n distinct eigenvalues) Suppose A is diagonalizable with independent eigenvectors v,, v n and corresponding eigenvalues λ,,λ n Let S be the matrix whose columns are the eigenvectors: S v v v n 3
4 Then As I observed above, On the other hand, since (S AS) n S A n S, Hence, e Dt λ λ S AS D λ n e λ t e Dt e λ t e λ nt ( t n (S AS) n ) S t n A n S S e At S n n e λ t e At e S λ t S e λ nt I can use this approach to compute e At in case A is diagonalizable Example Compute e At if A 3 5 The eigenvalues are λ, λ Since there are two different eigenvalues and A is a matrix, A is diagonalizable The corresponding eigenvectors are (5, ) and (, ) Thus, S 5, S 6 5 Hence, e At 5 e 4t e t ( 6 ) 5 6 5e 4t +e t 5e 4t 5e t e 4t e t e 4t +5e t Example Compute e At if A The eigenvalues are λ and λ (double) The corresponding eigenvectors are (3,,3) for λ, and (,,) and (,,) for λ Since I have 3 independent eigenvectors, the matrix is diagonalizable I have S 3, S
5 From this, it follows that e At 3et +4e t 6e t 6e t 6e t 6e t e t e t e t +3e t e t +e t 3e t +3e t 6e t 6e t 6e t 5e t Here s a quick check on the computation: If you set t in the right side, you get This checks, since e A I Note that this check isn t foolproof just because you get I by setting t doesn t mean your answer is right However, if you don t get I, your answer is surely wrong! How do you compute e At is A is not diagonalizable? I ll describe an iterative algorithm for computing e At that only requires that one know the eigenvalues of A There are various algorithms for computing the matrix exponential; this one, which is due to Williamson, seems to me to be the easiest for hand computation (Note that finding the eigenvalues of a matrix is, in general, a difficult problem: Any method for finding e At will have to deal with it) Let A be an n n matrix Let {λ,λ,,λ n } be a list of the eigenvalues, with multiple eigenvalues repeated according to their multiplicity Let a e λt, a k e λ kt a k (t) e λ k(t u) a k (u)du, k,,n, Then B I, B k (A λ k I) B k, k,,n, e At a B +a B ++a n B n To prove this, I ll show that the expression on the right satisfies the differential equation x A x To do this, I ll need two facts about the characteristic polynomial p(x) (x λ )(x λ ) (x λ n ) ±p(x) (Cayley-Hamilton Theorem) p(a) Observe that if p(x) is the characteristic polynomial, then using the first fact and the definition of the B s, p(x) ±(x λ )(x λ ) (x λ n ) p(a) ±(A λ I)(A λ I) (A λ n I) ±I(A λ I)(A λ I) (A λ n I) ±B (A λ I)(A λ I) (A λ n I) ±B (A λ I) (A λ n I) ±B n (A λ n I) 5
6 By the Cayley-Hamilton Theorem, ±B n (A λ n I) ( ) I will use this fact in the proof below Example I ll illustrate the Cayley-Hamilton theorem with the matrix A 3 The characteristic polynomial is ( λ)( λ) 6 λ 3λ 4 The Cayley-Hamilton theorem asserts that if you plug A into λ 3λ 4, you ll get the zero matrix First, A A A 4I Proof of the algorithm First, a k e λ k(t u) a k (u)du e λ kt Recall that the Fundamental Theorem of Calculus says that d dt f(u)du f(t) e λ ku a k (u)du Applying this and the Product Rule, I can differentiate a k to obtain a k λ k e λ kt e λ ku a k (u)du+e λ kt e λ kt a k (t), a k λ k a k +a k (a B +a B ++a n B n ) λ a B + λ a B +a B + λ 3 a 3 B 3 +a B 3 + λ n a n B n +a n B n Expand the a i B i terms using a i B i a i (A λ i I)B i a i AB i λ i a i B i 6
7 Making this substitution and telescoping the sum, I have λ a B + λ a B +a AB λ a B + λ 3 a 3 B 3 +a AB λ a B + λ n a n B n +a n AB n λ n a n B n λ n a n B n +A(a B +a B ++a n B n ) λ n a n B n Aa n B n +A(a B +a B ++a n B n ) a n (A λ n I)B n +A(a B +a B ++a n B n ) a n +A(a B +a B ++a n B n ) A(a B +a B ++a n B n ) (The result (*) proved above was used in the next-to-the-last equality) Combining the results above, I ve shown that (a B +a B ++a n B n ) A(a B +a B ++a n B n ) This shows that M a B +a B ++a n B n satisfies M AM Using the power series expansion, I have e ta A Ae ta So (e ta M) Ae ta M +e ta AM e ta AM +e ta AM (Remember that matrix multiplication is not commutative in general!) It follows that e ta M is a constant matrix Set t Since a a n, it follows that M() I In addition, e A I e ta M I, and hence M e At Example Use the matrix exponential to solve x 3 x, x() 3 4 The characteristic polynomial is (λ ) You can check that there is only one independent eigenvector, so I can t solve the system by diagonalizing I could use generalized eigenvectors to solve the system, but I will use the matrix exponential to illustrate the algorithm First, list the eigenvalues: {,} Since λ is a double root, it is listed twice First, I ll compute the a k s: a e t, Here are the B k s: a e t a (t) e (t u) e u du e t du te t B I, B (A I)B A I e At e t +te t e t +te t te t te t e t te t As a check, note that setting t produces the identity) 7
8 The solution to the given initial value problem is e x t +te t te t te t e t te t 3 4 You can get the general solution by replacing (3,4) with (c,c ) Example Find e At if A The eigenvalues are obviously λ (double) and λ First, I ll compute the a k s I have a e t, and a a 3 Next, I ll compute the B k s B I, and e t u e u du e t du te t, e (t u) ue u du te t e t +e t B A I, B 3 (A I)B e t e At te t e t te t +e t e t e t e t e t Example Use the matrix exponential to solve x 5 x 4 This example will demonstrate how the algorithm for e At works when the eigenvalues are complex The characteristic polynomial is λ + λ + The eigenvalues are λ ± i I will list them as { +i, i} First, I ll compute the a k s a e ( +i)t, and a e ( +i)(t u) e ( i)u du e ( +i)t e ( i)u e ( i)u du e ( +i)t e iu du e ( +i)ti ( e it ) i ( e ( i)t e ( +i)t) 8
9 Next, I ll compute the B k s B I, and B A ( +i)i 3 i 5 3 i e At e ( +i)t + i ( e ( i)t e ( +i)t) 3 i I want a real solution, so I ll use DeMoivre s Formula to simplify: e ( +i)t e t (cost+isint) 5 3 i e ( i)t e ( +i)t e t (cost isint) e t (cost+isint) ie t sint i ( e ( i)t e ( +i)t) e t sint Plugging these into the expression for e At above, I have e At e t (cost+isint) +e t 3 i 5 sint 3 i e t cost+3sint 5sint sint cost 3sint Notice that all the i s have dropped out! This reflects the obvious fact that the exponential of a real matrix must be a real matrix Finally, the general solution to the original system is x y e t cost+3sint 5sint sint cost 3sint c c Example I ll compare the matrix exponential and the eigenvector solution methods by solving the following system both ways: x x The characteristic polynomial is λ 4λ+5 The eigenvalues are λ ±i Consider λ +i: i A (+i)i i As this is an eigenvector matrix, it must be singular, and hence the rows must be multiples So ignore the second row I want a vector (a,b) such that ( i)a+( )b To get such a vector, switch the i and and negate one of them: a, b i Thus, (, i) is an eigenvector The corresponding solution is e (+i)t e t cost+isint i sint icost Take the real and imaginary parts: ree (+i)t i ime (+i)t i e t cost, sint e t sint cost 9
10 The solution is ( ) x e t cost sint c +c sint cost Now I ll solve the equation using the exponential The eigenvalues are {+i, i} Compute the a k s a e (+i)t, and a e ( i)t e (+i)t e ( i)(t u) e (+i)u du e ( i)t e iu du e ( i)t i eiu t i e( i)t( e it) i et( e it e it) e t sint (Here and below, I m cheating a little in the comparison by not showing all the algebra involved in the simplification You need to use DeMoivre s Formula to eliminate the complex exponentials) Next, compute the B k s B I, and i B A (+i)i i The solution is e At e (+i)t +e t i sint e t cost sint i sint cost x e t cost sint sint cost Taking into account some of the algebra I didn t show for the matrix exponential, I think the eigenvector approach is easier c c Example Solve the system x 5 8 x 3 For comparison, I ll do this first using the generalized eigenvector method, then using the matrix exponential The characteristic polynomial is λ λ+ The eigenvalue is λ (double) A I Ignore the first row, and divide the second row by, obtaining the vector (, ) I want (a,b) such that ()a+( )b Swap and and negate the : I get (a,b) (,) This is an eigenvector for λ Since I only have one eigenvector, I need a generalized eigenvector This means I need (a,b ) such that 4 8 a 4 b Row reduce: This means that a b + Setting b yields a The generalized eigenvector is (, )
11 The solution is ( ) x c e t +c te t +e t Next, I ll solve the system using the matrix exponential The eigenvalues are {, } First, I ll compute the a k s a e t, and a e t e t e t u e u du e t du te t Next, compute the B k s B I, and The solution is B A I e At e t +te t 4 8 e t +4te t 8te t 4 te t e t 4te t e x t +4te t 8te t te t e t 4te t c In this case, finding the solution using the matrix exponential may be a little bit easier c Richard Williamson, Introduction to differential equations Englewood Cliffs, NJ: Prentice-Hall, 986 c by Bruce Ikenaga
MA2327, Problem set #3 (Practice problems with solutions)
MA2327, Problem set #3 (Practice problems with solutions) 5 2 Compute the matrix exponential e ta in the case that A = 2 5 2 Compute the matrix exponential e ta in the case that A = 5 3 Find the unique
More informationSystems of Linear ODEs
P a g e 1 Systems of Linear ODEs Systems of ordinary differential equations can be solved in much the same way as discrete dynamical systems if the differential equations are linear. We will focus here
More informationPartial Fractions. (Do you see how to work it out? Substitute u = ax + b, so du = a dx.) For example, 1 dx = ln x 7 + C, x x (x 3)(x + 1) = a
Partial Fractions 7-9-005 Partial fractions is the opposite of adding fractions over a common denominator. It applies to integrals of the form P(x) dx, wherep(x) and Q(x) are polynomials. Q(x) The idea
More information[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]
Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the
More informationSystems of differential equations Handout
Systems of differential equations Handout Peyam Tabrizian Friday, November 8th, This handout is meant to give you a couple more example of all the techniques discussed in chapter 9, to counterbalance all
More informationHomogeneous Linear Systems of Differential Equations with Constant Coefficients
Objective: Solve Homogeneous Linear Systems of Differential Equations with Constant Coefficients dx a x + a 2 x 2 + + a n x n, dx 2 a 2x + a 22 x 2 + + a 2n x n,. dx n = a n x + a n2 x 2 + + a nn x n.
More informationLinear Algebra Basics
Linear Algebra Basics For the next chapter, understanding matrices and how to do computations with them will be crucial. So, a good first place to start is perhaps What is a matrix? A matrix A is an array
More informationMatrix-Exponentials. September 7, dx dt = ax. x(t) = e at x(0)
Matrix-Exponentials September 7, 207 In [4]: using PyPlot INFO: Recompiling stale cache file /Users/stevenj/.julia/lib/v0.5/LaTeXStrings.ji for module LaTeXString Review: Solving ODEs via eigenvectors
More informationSystems 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 informationCalculus II. Calculus II tends to be a very difficult course for many students. There are many reasons for this.
Preface Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus
More informationMa 227 Review for Systems of DEs
Ma 7 Review for Systems of DEs Matrices Basic Properties Addition and subtraction: Let A a ij mn and B b ij mn.then A B a ij b ij mn 3 A 6 B 6 4 7 6 A B 6 4 3 7 6 6 7 3 Scaler Multiplication: Let k be
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 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationx + 2y + 3z = 8 x + 3y = 7 x + 2z = 3
Chapter 2: Solving Linear Equations 23 Elimination Using Matrices As we saw in the presentation, we can use elimination to make a system of linear equations into an upper triangular system that is easy
More informationGeneralized eigenspaces
Generalized eigenspaces November 30, 2012 Contents 1 Introduction 1 2 Polynomials 2 3 Calculating the characteristic polynomial 5 4 Projections 7 5 Generalized eigenvalues 10 6 Eigenpolynomials 15 1 Introduction
More informationMath 215 HW #11 Solutions
Math 215 HW #11 Solutions 1 Problem 556 Find the lengths and the inner product of 2 x and y [ 2 + ] Answer: First, x 2 x H x [2 + ] 2 (4 + 16) + 16 36, so x 6 Likewise, so y 6 Finally, x, y x H y [2 +
More informationa 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 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 informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationName Solutions Linear Algebra; Test 3. Throughout the test simplify all answers except where stated otherwise.
Name Solutions Linear Algebra; Test 3 Throughout the test simplify all answers except where stated otherwise. 1) Find the following: (10 points) ( ) Or note that so the rows are linearly independent, so
More informationLinear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions
Linear Algebra, part 2 Eigenvalues, eigenvectors and least squares solutions Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Main problem of linear algebra 2: Given
More informationCommutative Rings and Fields
Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two
More informationRecitation 9: Probability Matrices and Real Symmetric Matrices. 3 Probability Matrices: Definitions and Examples
Math b TA: Padraic Bartlett Recitation 9: Probability Matrices and Real Symmetric Matrices Week 9 Caltech 20 Random Question Show that + + + + +... = ϕ, the golden ratio, which is = + 5. 2 2 Homework comments
More informationMIT 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 informationLINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. General framework
Differential Equations Grinshpan LINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. We consider linear ODE of order n: General framework (1) x (n) (t) + P n 1 (t)x (n 1) (t) + + P 1 (t)x (t) + P 0 (t)x(t) = 0
More informationGetting Started with Communications Engineering
1 Linear algebra is the algebra of linear equations: the term linear being used in the same sense as in linear functions, such as: which is the equation of a straight line. y ax c (0.1) Of course, if we
More informationSums and Products. a i = a 1. i=1. a i = a i a n. n 1
Sums and Products -27-209 In this section, I ll review the notation for sums and products Addition and multiplication are binary operations: They operate on two numbers at a time If you want to add or
More information1300 Linear Algebra and Vector Geometry
1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Matrix Inversion Algorithm One payoff from this theorem: It gives us a way to invert matrices.
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 310, REVIEW SHEET 2
MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,
More informationMath 304 Answers to Selected Problems
Math Answers to Selected Problems Section 6.. Find the general solution to each of the following systems. a y y + y y y + y e y y y y y + y f y y + y y y + 6y y y + y Answer: a This is a system of the
More informationWe have already seen that the main problem of linear algebra is solving systems of linear equations.
Notes on Eigenvalues and Eigenvectors by Arunas Rudvalis Definition 1: Given a linear transformation T : R n R n a non-zero vector v in R n is called an eigenvector of T if Tv = λv for some real number
More informationAdvanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. #07 Jordan Canonical Form Cayley Hamilton Theorem (Refer Slide Time:
More informationGetting Started with Communications Engineering. Rows first, columns second. Remember that. R then C. 1
1 Rows first, columns second. Remember that. R then C. 1 A matrix is a set of real or complex numbers arranged in a rectangular array. They can be any size and shape (provided they are rectangular). A
More informationDifferential equations
Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the x-component of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system
More informationLinear Algebra, Summer 2011, pt. 2
Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................
More informationMatrices. 1 a a2 1 b b 2 1 c c π
Matrices 2-3-207 A matrix is a rectangular array of numbers: 2 π 4 37 42 0 3 a a2 b b 2 c c 2 Actually, the entries can be more general than numbers, but you can think of the entries as numbers to start
More informationMAT 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 informationMaths for Signals and Systems Linear Algebra in Engineering
Maths for Signals and Systems Linear Algebra in Engineering Lectures 13-14, Tuesday 11 th November 2014 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON Eigenvectors
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 informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationLinear Algebra Review (Course Notes for Math 308H - Spring 2016)
Linear Algebra Review (Course Notes for Math 308H - Spring 2016) Dr. Michael S. Pilant February 12, 2016 1 Background: We begin with one of the most fundamental notions in R 2, distance. Letting (x 1,
More informationπ 1 = tr(a), π n = ( 1) n det(a). In particular, when n = 2 one has
Eigen Methods Math 246, Spring 2009, Professor David Levermore Eigenpairs Let A be a real n n matrix A number λ possibly complex is an eigenvalue of A if there exists a nonzero vector v possibly complex
More informationGroups. s t or s t or even st rather than f(s,t).
Groups Definition. A binary operation on a set S is a function which takes a pair of elements s,t S and produces another element f(s,t) S. That is, a binary operation is a function f : S S S. Binary operations
More informationPartial Fractions. (Do you see how to work it out? Substitute u = ax+b, so du = adx.) For example, 1 dx = ln x 7 +C, x 7
Partial Fractions -4-209 Partial fractions is the opposite of adding fractions over a common denominator. It applies to integrals of the form P(x) dx, wherep(x) and Q(x) are polynomials. Q(x) The idea
More informationMath 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 informationEcon Lecture 14. Outline
Econ 204 2010 Lecture 14 Outline 1. Differential Equations and Solutions 2. Existence and Uniqueness of Solutions 3. Autonomous Differential Equations 4. Complex Exponentials 5. Linear Differential Equations
More informationThe goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T
1 1 Linear Systems The goal of this chapter is to study linear systems of ordinary differential equations: ẋ = Ax, x(0) = x 0, (1) where x R n, A is an n n matrix and ẋ = dx ( dt = dx1 dt,..., dx ) T n.
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationMatrix Exponential Formulas
Matrix Exponential Formulas Linear Analysis May 3, 016 Abstract We present some tricks for quickly calculating the matrix exponential of certain special classes of matrices, such as matrices. 1 matrix
More information18.06 Problem Set 8 - Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8 - Solutions Due Wednesday, 4 November 2007 at 4 pm in 2-06 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
More informationExamples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling
1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples
More informationDifferential Equations
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
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 informationSections 6.1 and 6.2: Systems of Linear Equations
What is a linear equation? Sections 6.1 and 6.2: Systems of Linear Equations We are now going to discuss solving systems of two or more linear equations with two variables. Recall that solving an equation
More informationExamples of Groups
Examples of Groups 8-23-2016 In this section, I ll look at some additional examples of groups. Some of these will be discussed in more detail later on. In many of these examples, I ll assume familiar things
More informationLecture 2j Inner Product Spaces (pages )
Lecture 2j Inner Product Spaces (pages 348-350) So far, we have taken the essential properties of R n and used them to form general vector spaces, and now we have taken the essential properties of the
More informationNovember 18, 2013 ANALYTIC FUNCTIONAL CALCULUS
November 8, 203 ANALYTIC FUNCTIONAL CALCULUS RODICA D. COSTIN Contents. The spectral projection theorem. Functional calculus 2.. The spectral projection theorem for self-adjoint matrices 2.2. The spectral
More informationDIFFERENTIAL EQUATIONS REVIEW. Here are notes to special make-up discussion 35 on November 21, in case you couldn t make it.
DIFFERENTIAL EQUATIONS REVIEW PEYAM RYAN TABRIZIAN Here are notes to special make-up discussion 35 on November 21, in case you couldn t make it. Welcome to the special Friday after-school special of That
More informationNotes on the Matrix-Tree theorem and Cayley s tree enumerator
Notes on the Matrix-Tree theorem and Cayley s tree enumerator 1 Cayley s tree enumerator Recall that the degree of a vertex in a tree (or in any graph) is the number of edges emanating from it We will
More informationREVIEW PROBLEMS FOR MIDTERM II MATH 2373, FALL 2016 ANSWER KEY
REVIEW PROBLEMS FOR MIDTERM II MATH 7, FALL 6 ANSWER KEY This list of problems is not guaranteed to be an absolutely complete review. For completeness you must also make sure that you know how to do all
More informationDifferential Equations and Linear Algebra - Fall 2017
NAME: Differential Equations and Linear Algebra - Fall 27 Final Exam, December 4, 27 GRADING:. In multiple choice problems -3 you don t have to show your work. Consequently, no partial credit will be given.
More informationEXAMPLES OF PROOFS BY INDUCTION
EXAMPLES OF PROOFS BY INDUCTION KEITH CONRAD 1. Introduction In this handout we illustrate proofs by induction from several areas of mathematics: linear algebra, polynomial algebra, and calculus. Becoming
More informationInverses and Elementary Matrices
Inverses and Elementary Matrices 1-12-2013 Matrix inversion gives a method for solving some systems of equations Suppose 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 n1 x
More information22A-2 SUMMER 2014 LECTURE 5
A- SUMMER 0 LECTURE 5 NATHANIEL GALLUP Agenda Elimination to the identity matrix Inverse matrices LU factorization Elimination to the identity matrix Previously, we have used elimination to get a system
More informationComplex Numbers and Exponentials
Complex Numbers and Exponentials Definition and Basic Operations A complexnumber is nothing morethan a point in the xy plane. The first component, x, ofthe complex number (x,y) is called its real part
More information6.4 Division of Polynomials. (Long Division and Synthetic Division)
6.4 Division of Polynomials (Long Division and Synthetic Division) When we combine fractions that have a common denominator, we just add or subtract the numerators and then keep the common denominator
More informationFirst we introduce the sets that are going to serve as the generalizations of the scalars.
Contents 1 Fields...................................... 2 2 Vector spaces.................................. 4 3 Matrices..................................... 7 4 Linear systems and matrices..........................
More informationsystems of linear di erential If the homogeneous linear di erential system is diagonalizable,
G. NAGY ODE October, 8.. Homogeneous Linear Differential Systems Section Objective(s): Linear Di erential Systems. Diagonalizable Systems. Real Distinct Eigenvalues. Complex Eigenvalues. Repeated Eigenvalues.
More informationLinear Algebra Primer
Introduction Linear Algebra Primer Daniel S. Stutts, Ph.D. Original Edition: 2/99 Current Edition: 4//4 This primer was written to provide a brief overview of the main concepts and methods in elementary
More informationThe eigenvalues are the roots of the characteristic polynomial, det(a λi). We can compute
A. [ 3. Let A = 5 5 ]. Find all (complex) eigenvalues and eigenvectors of The eigenvalues are the roots of the characteristic polynomial, det(a λi). We can compute 3 λ A λi =, 5 5 λ from which det(a λi)
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
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 informationLinear Algebra 2 Spectral Notes
Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex
More informationBasic Linear Algebra in MATLAB
Basic Linear Algebra in MATLAB 9.29 Optional Lecture 2 In the last optional lecture we learned the the basic type in MATLAB is a matrix of double precision floating point numbers. You learned a number
More informationPhysics 505 Homework No. 1 Solutions S1-1
Physics 505 Homework No s S- Some Preliminaries Assume A and B are Hermitian operators (a) Show that (AB) B A dx φ ABψ dx (A φ) Bψ dx (B (A φ)) ψ dx (B A φ) ψ End (b) Show that AB [A, B]/2+{A, B}/2 where
More informationPractice Final Exam Solutions for Calculus II, Math 1502, December 5, 2013
Practice Final Exam Solutions for Calculus II, Math 5, December 5, 3 Name: Section: Name of TA: This test is to be taken without calculators and notes of any sorts. The allowed time is hours and 5 minutes.
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More 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 informationc 1 v 1 + c 2 v 2 = 0 c 1 λ 1 v 1 + c 2 λ 1 v 2 = 0
LECTURE LECTURE 2 0. Distinct eigenvalues I haven t gotten around to stating the following important theorem: Theorem: A matrix with n distinct eigenvalues is diagonalizable. Proof (Sketch) Suppose n =
More informationSAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra
1.1. Introduction SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that
More informationES.1803 Topic 19 Notes Jeremy Orloff. 19 Variation of parameters; exponential inputs; Euler s method
ES83 Topic 9 Notes Jeremy Orloff 9 Variation of parameters; eponential inputs; Euler s method 9 Goals Be able to derive and apply the eponential response formula for constant coefficient linear systems
More informationMath Spring 2011 Final Exam
Math 471 - Spring 211 Final Exam Instructions The following exam consists of three problems, each with multiple parts. There are 15 points available on the exam. The highest possible score is 125. Your
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationTHE MINIMAL POLYNOMIAL AND SOME APPLICATIONS
THE MINIMAL POLYNOMIAL AND SOME APPLICATIONS KEITH CONRAD. Introduction The easiest matrices to compute with are the diagonal ones. The sum and product of diagonal matrices can be computed componentwise
More informationTopic 15 Notes Jeremy Orloff
Topic 5 Notes Jeremy Orloff 5 Transpose, Inverse, Determinant 5. Goals. Know the definition and be able to compute the inverse of any square matrix using row operations. 2. Know the properties of inverses.
More informationTopics in linear algebra
Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over
More informationLinear Algebra. The Manga Guide. Supplemental Appendixes. Shin Takahashi, Iroha Inoue, and Trend-Pro Co., Ltd.
The Manga Guide to Linear Algebra Supplemental Appendixes Shin Takahashi, Iroha Inoue, and Trend-Pro Co., Ltd. Copyright by Shin Takahashi and TREND-PRO Co., Ltd. ISBN-: 978--97--9 Contents A Workbook...
More information21 Linear State-Space Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More informationProblems for M 11/2: A =
Math 30 Lesieutre Problem set # November 0 Problems for M /: 4 Let B be the basis given by b b Find the B-matrix for the transformation T : R R given by x Ax where 3 4 A (This just means the matrix for
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 informationDivisibility = 16, = 9, = 2, = 5. (Negative!)
Divisibility 1-17-2018 You probably know that division can be defined in terms of multiplication. If m and n are integers, m divides n if n = mk for some integer k. In this section, I ll look at properties
More informationMATH 583A REVIEW SESSION #1
MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),
More informationA Brief Outline of Math 355
A Brief Outline of Math 355 Lecture 1 The geometry of linear equations; elimination with matrices A system of m linear equations with n unknowns can be thought of geometrically as m hyperplanes intersecting
More informationLinear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More 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 informationMATHEMATICS 217 NOTES
MATHEMATICS 27 NOTES PART I THE JORDAN CANONICAL FORM The characteristic polynomial of an n n matrix A is the polynomial χ A (λ) = det(λi A), a monic polynomial of degree n; a monic polynomial in the variable
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 informationLinear Algebra (MATH ) Spring 2011 Final Exam Practice Problem Solutions
Linear Algebra (MATH 4) Spring 2 Final Exam Practice Problem Solutions Instructions: Try the following on your own, then use the book and notes where you need help. Afterwards, check your solutions with
More information= AΦ, Φ(0) = I, = k! tk A k = I + ta t2 A t3 A t4 A 4 +, (3) e ta 1
Matrix Exponentials Math 26, Fall 200, Professor David Levermore We now consider the homogeneous constant coefficient, vector-valued initial-value problem dx (1) = Ax, x(t I) = x I, where A is a constant
More information