Week 13: 23 Nov, |
24 Nov, | 25 Nov, |

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. Slides and problem notes exist for 6.1 and 6.2 problems. See the web site.Google AlgorithmLawrence Page's pagerank algorithm, google web page rankings. Relationship to the power method, stochastic matrices and eigenanalysis.

Algebraic Eigenanalysis Section 6.2.Calculation of eigenpairs to produce Fourier's model. Connection between Fourier's model and a diagonalizable matrix. Further calculation examples. How to deal with examples where A has an eigenvalue of multiplicity greater than one.Cayley-Hamilton topics.Computing powers of matrices. Stochastic matrices. Example of 1984 telecom companies MCI, SPRINT, ATT with discrete dynamical system u(n+1)=A u(n). Matrix A is stochastic.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 + 3y, y' = 4y, x(0)=1, y(0)=2. Find the eigenpairs (2, v1), (4,v2) where v1=vector(1,0]) and v2=vector([3,2]). 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 ] [ 3 ] u(t) = c1 e^{2t} [ ] + c2 e^4t} [ ] [ 0 ] [ 2 ] which means x(t) = c1 exp(2t) + 3 c2 exp(4t), y(t) = 2 c2 exp(4t).

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([[2,3],[0,4]]). Dynamical system scalar form is x' = 2x + 3y, y' = 4y, x(0)=1, y(0)=2. Laplace resolvent method The shortcut equations Solving for L(x), L(y) Backward table and Lerch's theorem Answers for x(t), y(t) Chapter 1+5 Method Solve w'+p(t)w=0 as w = constant / integrating factor. Then y(t) = 2 exp(4t) Stuff y(t) into the first DE to get the linear DE x' - 2x = 6 exp(4t) Superposition: x(t)=x_h(t)+x_p(t), x_h(t)=c exp(2t), x_p(t) = d1 exp(4t) = 3 exp(4t) by undetermined coeff. Then x(t)= -2 exp(2t) + 3 exp(4t).Cayley-Hamilton 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 Ziebur's Lemma the 2x2 dynamical system above. The characteristic equation is (2-lambda)(4-lambda)=0 with roots lambda = 2,4 Euler's theorem implies the atoms are exp(2t), exp(4t). Ziebur's Lemma says that u(t) = exp(2t) u_1 + exp(4t) u_2 where vectors u_1, U_2 are to be determined from A and the initial conditions x(0)=1, y(0)=2. ZIEBUR ALGORITHM. To solve for u_1, u_2 in the example, differentiate the equation and set t=0 in both relations. Then u'=Au implies u_0 = u_1 + u_2, Au_0 = 2 u_1 + 4 u_2. These equations can be solved by elimination. The answer: u_1 = -(Au_0 - 4 u_0)/2, u_2 = (Au_0 - 2 u_0)/2 = vector([-2,0]) = vector([3,2]) These are recognized as eigenvectors of A for lambda=2 and lambda=4, respectively. ZIEBUR SHORTCUT [a textbook method] Start with Ziebur's theorem, which implies that x(t) = k1 exp(2t) + k2 exp(4t). Use the first DE to solve for y(t): y(t) = (1/3)(x'(t) - 2x(t)) = (1/3)(2 k1 exp(2t) + 4 k2 exp(4t) - 2 k1 exp(2t) - 2 k2 exp(4t)) = (2/3) k2 exp(4t) For example, x(0)=1, y(0)=2 implies k1 and k2 are defined by k1 + k2 = 1, (2/3) k2 = 2, which implies k1 = -2, k2 = 3, agreeing with a previous solution formula.Survey of Methods for solving a 2x2 dynamical system1. Cayley-Hamilton method for u'=Au Solution: u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n) Atoms: They are constructed by Euler's theorem from roots of det(A-rI)=0 Vectors: Symbols vec(c_1), ..., vec(c_n) are not arbitrary. They are determined from A and u(0). Algorithm outlined above for 2x2. 2. Laplace resolvent L(u)=(s I - A)^(-1) u(0) 3. Eigenanalysis u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2 4. Putzer's method for the 2x2 matrix exponential. Solution of u'=Au is: u(t) = exp(A t)u(0) THEOREM: exp(A t) = r1(t) I + r2(t) (A-lambda_1 I), Lambda Symbols: lambda_1 and lambda_2 are the roots of det(A-lambda I)=0. The DE System: r1'(t) = lambda_1 r1(t), r1(0)=0, r2'(t) = lambda_2 r2(t) + r1(t), r2(0)=0Topics from linear systems:Brine tank models. Recirculating brine tanks. Pond pollution. Home heating. Earthquakes. Railway cars. All are 2x2 or 3x3 or nxn system applications that can be solved by Laplace methods.

Engineering modelsThe job-site cable hoist example [delayed] Sliding plates example [delayed] Home heating example [more coming]