Topics from chapter 10, EPbvp7.6. Problems 1,2 are on Laplace theory, but there is some contact with chapter 5, sections 5.1 to 5.4. See the online sample exam for details. Problems 3,4,5 will be on April 19.Sample Exam 3 for S2012: sample exam 3, all five problems. (182.4 K, pdf, 14 Nov 2010)

REVIEW from Last WeekConversion Methods to Create a First Order SystemThe position-velocity substitution. How to convert second order systems. How to convert nth order scalar differential equations. Non-homogeneous terms and the vector matrix system u' = Au + F(t) Non-linear systems and the vector-matrix system u' = F(t,u) Answer checks for u'=Au Example: The system u'=Au, A=matrix([[2,1],[0,3]]);Systems of two differential equationsThe Laplace resolvent method for systems. Solving the resolvent equation for L(x), L(y). Cramer's Rule Matrix inversion Elimination Example: Solving a 2x2 dynamical system Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,1],[0,3]]). Dynamical system scalar form is x' = 2x + y, y' = 3y, x(0)=1, y(0)=2. The equations for L(x), L(y) (s-2)L(x) + (-1)L(y)=1, (0)L(x) + (s-3)L(y)=2 REMARK: Laplace resolvent method shortcut. How to solve the [resolvent] equations for L(x), L(y). Cramer's Rule Matrix inversion Elimination Answers: L(x) = delta1/delta, L(y)=delta2/delta delta=(s-2)(s-3), delta1=s-1, delta2=2(s-2) L(x) = -1/(s-2)+2/(s-3), L(y)=2/(s-3) Backward table and Lerch's theorem Answers: x(t) = - e^{2t} + 2 e^{3t}, y(t) = 2 e^{3t}. Edwards-Penney Shortcut Method in Example 5, 7.1. Uses Chapter 1+5 methods. This is the Cayley-Hamilton-Ziebur method. See below. Solve w'+p(t)w=0 as w = constant / integrating factor. Then y' -2y=0 ==> y(t) = 2 exp(3t) Stuff y(t) into the first DE to get the linear DE x' - 2x = 2 exp(3t) Superposition: x(t)=x_h(t)+x_p(t), x_h(t)=c exp(2t), x_p(t) = d1 exp(t) = 2 exp(3t) by undetermined coeff. Then x(t)= - exp(2t) + 2 exp(3t).Cayley-Hamilton TheoremA matrix satisfies its own characteristic equation. ILLUSTRATION: det(A-r I)=0 for the previous example is (2-r)(3-r)=0 or r^2 -5r + 6=0. Then C-H says A^2 - 5A + 6I = 0.Cayley-Hamilton-Ziebur MethodZIEBUR'S LEMMA. The components of u in u'=Au are linear combinations of the atoms created by Euler's theorem applied to the roots of the characteristic equation det(A-rI)=0. THEOREM. Solve u'=Au without complex numbers or eigenanalysis. The solution of u'=Au is a linear combination of atoms times certain constant vectors [not arbitrary vectors]. u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n) PROBLEM: Solve by Cayley-Hamilton-Ziebur the 2x2 dynamical system x' = 2x + y, y' = 3y, x(0)=1, y(0)=2. The characteristic equation is (2-lambda)(3-lambda)=0 with roots lambda = 2,3 Euler's theorem implies the atoms are exp(2t), exp(3t). Ziebur's Theorem says that u(t) = exp(2t) vec(u_1) + exp(3t) vec(u_2) where vectors u_1, u_2 are to be determined from the matrix A = matrix([[2,1],[0,3]]) and initial conditions x(0)=1, y(0)=2. ZIEBUR ALGORITHM. To solve for u_1, u_2 in the example, differentiate the equation u(t) = exp(2t) u_1 + exp(3t) u_2 and set t=0 in both relations. Then u'=Au implies u_0 = u_1 + u_2, Au_0 = 2 u_1 + 3 u_2. These equations can be solved by elimination. The answer: u_1 = (3 u_0 -Au_0), u_2 = (Au_0 - 2 u_0) = vector([-1,0]) = vector([2,2]) Vectors u_1, u_2 are recognized as eigenvectors of A for lambda=2 and lambda=3, respectively, after studying chapter 6. ZIEBUR SHORTCUT [Edwards-Penney textbook method, Example 5 in 7.1] Start with Ziebur's theorem, which implies that x(t) = k1 exp(2t) + k2 exp(3t). Use the first DE to solve for y(t): y(t) = x'(t) - 2x(t) = 2 k1 exp(2t) + 3 k2 exp(3t) - 2 k1 exp(2t) - 2 k2 exp(3t)) = k2 exp(3t) For example, x(0)=1, y(0)=2 implies k1 and k2 are defined by k1 + k2 = 1, k2 = 2, which implies k1 = -1, k2 = 2, agreeing with a previous solution formula.

EIGENANALYSIS WARNINGReading Edwards-Penney Chapter 6 may deliver the wrong ideas about how to solve for eigenpairs. The examples emphasize a clever shortcut, which does not apply in general to solve for eigenpairs. HISTORY. Chapter 6 originally appeared in the 2280 book as a summary, which assumed a linear algebra course. The chapter was copied without changes into the Edwards-Penney Differential Equations and Linear Algebra textbook, which you currently own. The text contains only shortcuts. There is no discussion of a general method for finding eigenpairs. You will have to fill in the details by yourself. The online lecture notes and slides were created to fill in the gap.Lecture: Fourier's Model. Intro to Eigenanalysis, Ch6.Examples and motivation. Ellipse, rotations, eigenpairs. General solution of a differential equation u'=Au and eigenpairs. Fourier's model. History. J.B.Fourier's 1822 treatise on the theory of heat. The rod example. Physical Rod: a welding rod of unit length, insulated on the lateral surface and ice packed on the ends. Define f(x)=thermometer reading at loc=x along the rod at t=0. Define u(x,t)=thermometer reading at loc=x and time=t>0. Problem: Find u(x,t). Fourier's solution assume that f(x) = 17 sin (pi x) + 29 sin(5 pi x) = 17 v1 + 29 v2 Packages v1, v2 are vectors in a vector space V of functions on [0,1]. Fourier computes u(x,t) by re-scaling v1, v2 with numbers Lambda_1, Lambda_2 that depend on t. This idea is calledFourier's Model.u(x,t) = 17 ( exp(-pi^2 t) sin(pi x)) + 29 ( exp(-25 pi^2 t) sin (5 pi x)) = 17 (Lambda_1 v1) + 29 (Lambda_2 v2) Eigenanalysis of u'=Au is the identical idea. u(0) = c1 v1 + c2 v2 implies u(t) = c1 exp(lambda_1 t) v1 + c2 exp(lambda_2 t) v2 Fourier's re-scaling idea from 1822, applied to u'=Au, replaces v1 and v2 in the expression c1 v1 + c2 v2 by their re-scaled versions to obtain the answer c1 (Lambda1 v1) + c2 (Lambda2 v2) where Lambda1 = exp(lambda_1 t), Lambda2 = exp(lambda_2 t).

Main Theorem on Fourier's ModelTHEOREM. Fourier's model A(c1 v1 + c2 v2) = c1 (lambda1 v1) + c2 (lambda2 v2) with v1, v2 a basis of R^2 holds [for all constants c1, c2] if and only if the vector-matrix system A(v1) = lambda1 v1, A(v2) = lambda2 v2, has a solution with vectors v1, v2 independent if and only if the diagonal matrix D=diag(lambda1,lambda2) and the augmented matrix P=aug(v1,v2) satisfy 1. det(P) not zero [then v1, v2 are independent] 2. AP=PD THEOREM. The eigenvalues of A are found from the determinant equation det(A -lambda I)=0, which is called the characteristic equation. THEOREM. The eigenvectors of A are found from the frame sequence which starts with B=A-lambda I [lambda a root of the characteristic equation], ending with last frame rref(B). The eigenvectors for lambda are the partial derivatives of the general solution obtained by the Last Frame Algorithm, with respect to the invented symbols t1, t2, t3, ...

Algebraic Eigenanalysis Section 6.2.Calculation of eigenpairs to produce Fourier's model. Connection between Fourier's model and a diagonalizable matrix. How to find the variables lambda and v in Fourier's model using determinants and frame sequences. Solved in class: examples similar to the problems in 6.1 and 6.2. Web slides and problem notes exist for the 6.1 and 6.2 problems. Examples where A has an eigenvalue of multiplicity greater than one.

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Eigenanalysis ExamplesProblems 6.1, 6.2. See also FAQ online.Solving DE System u' = Au by EigenanalysisExample: Solving a 2x2 dynamical system Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,3],[0,4]]). Dynamical system scalar form is x' = 2x + 1y, y' = 3y, x(0)=1, y(0)=2. Find the eigenpairs (2, v1), (3,v2) where v1=vector([1,0]) and v2=vector([1,1]). THEOREM. The solution of u' = Au in the 2x2 case is u(t) = c1 exp(lambda1 t) v1 + c2 exp(lambda2 t) v2 APPLICATION: u(t) = c1 exp(2t) v1 + c2 exp(4t) v2 [ 1 ] [ 1 ] u(t) = c1 e^{2t} [ ] + c2 e^4t} [ ] [ 0 ] [ 1 ] which means x(t) = c1 exp(2t) + 3 c2 exp(4t), y(t) = 2 c2 exp(4t).Diagonalization TheoryIn the case of a 2x2 matrix A, FOURIER'S MODEL is A(c1 v1 + c2 v2) = c1(lambda1 v1) + c2(lambda2 v2) where v1,v2 are a basis for the plane equivalent to DIAGONALIZATION AP=PD, where D=diag(lamba1,lambda2), P=augment(v1,v2), where det(P) is not zero equivalent to EIGENPAIR EQUATIONS A(v1)=lambda1 v1, A(v2)=lambda2 v2, where vectors v1,v2 are independentExamplesGiven the eigenpairs of A, find A via AP=PD. Given P, D, then find A. Given A, then find P, D.Cayley-Hamilton topics, Section 6.3.Computing powers of matrices. Stochastic matrices. Example of 1984 telecom companies ATT, MCI, SPRINT with discrete dynamical system u(n+1)=A u(n). Matrix A is stochastic. EXAMPLE: [ 6 1 5 ] [ a(t) ] 10 A = [ 2 7 1 ] u(t) = [ m(t) ] [ 2 2 4 ] [ s(t) ] Meaning: 60% stay with ATT and 20% switch to MCI, 20% switch to SPRINT. 70% stay with MCI and 20% switch to SPRINT, 10% switch to ATT. 40% stay with SPRINT and 50% switch to ATT, 10% switch to MCI. Powers of A and the meaning of A^n x_0 for the telecom example.Google AlgorithmLawrence Page's pagerank algorithm, google web page rankings. Eigenanalysis and powers of A.: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)Text