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 independentDrill Problems1. Problem: Given P and D, find A in the relation AP=PD. 2. Problem: Given Fourier's model, find A. 3. Problem: Given A, find Fourier's model. 4. Problem: Given A, find all eigenpairs. 5. Problem: Given A, find packages P and D such that AP=PD. 6. Problem: Give an example of a matrix A which has no Fourier's model. 7. Problem: Give an example of a matrix A which is not diagonalizable. 8. Problem: Given 2 eigenpairs, find the 2x2 matrix A.Cayley-Hamilton topics, Section 6.3.Power MethodComputing 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.Google AlgorithmLawrence Page's pagerank algorithm, google web page rankings.

Methods to solve dynamical systemsConsider the 2x2 system x'=x-5y, y'=x-y, x(0)=1, y(0)=2. Cayley-Hamilton-Ziebur method. Laplace resolvent. Eigenanalysis method. Exponential matrix using maple Putzer's method to compute the exponential matrix Spectral methods [ch8; not studied in 2250]

Survey of Methods for solving a 2x2 dynamical system1. Cayley-Hamilton-Ziebur 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). For the algorithm, see the slides. 2. Laplace resolvent L(u)=(s I - A)^(-1) u(0) See slides for details about the resolvent equation. 3. Eigenanalysis u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2 See chapter 7 in Edwards-Penney for examples and details. 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)=0 See the slides and manuscript on systems for proofs and details. THEOREM. The formula can be used as e^{r1 t} - e^{r2 t} e^{At} = e^{r1 t} I + ------------------- (A-r1 I) r1 - r2 where r1=lambda_1, r2=lambda_2 are the eigenavalues of A. EXAMPLE. Solve a homogeneous system u'=Au, u(0)=vector([1,2]), A=matrix([[2,3],[0,4]]) using the matrix exponential, Zeibur's method, Laplace resolvent and eigenanalysis. EXAMPLE. Solve a non-homogeneous system u'=Au+F(t), u(0)=vector([0,0]), A=matrix([[2,3],[0,4]]), F(t)=vector([3,1]) using variation of parameters.

Exam 3 ReviewShortest trial solution in undetermined coefficients. Example: Sample exam. Eigenvalues A 4x4 matrix. Block determinant theorem. Eigenvectors for a 4x4. B:=matrix([[5,0,0,0],[0,5,0,0],[0,0,0,3],[0,0,-3,0]]); lambda=5,5,3i,-3i v1=[1,0,0,0], v2=[0,1,0,0], v3=[0,0,i,-1], v4=[0,0,i,1] One panel for lambda=5 First frame is A-5I with 0 appended Find rref Apply last frame algorithm Scalar general solution Take partials on t1, t2 to find v1,v2 Eigenpairs are (5,v1), (5,v2) One panel for lambda=3i Same outline as lambda=5 Get one eigenpair (3i,v3) Other eigenpair=(-3i,v4) where v4 is the conjugate of v3. Final exam: Second shifting theorem in Laplace theory.Second Order SystemsHow to convert mx''+cx'+kx=F0 cos (omega t) into a dynamical system u'=Au+F(t). Electrical systems u'=Au+E(t) from LRC circuit equations. Electrical systems of order two: networks Mechanical systems of order two: coupled systems Second order systems u''=Au+F Examples are railway cars, earthquakes, vibrations of multi- component systems, electrical networks.Second Order Vector-Matrix Differential EquationsThe model u'' = Ax + F(t) Coupled Spring-Mass System. Problem 7.4-6 A:=matrix([[-6,4],[2,-4]]); eigenvals(A); lambda1= -2, lambda2= -8Ziebur's Methodroots for Ziebur's theorem are plus or minus sqrt(lambda) Roots = sqrt(2)i, sqrt(8)i, -sqrt(2)i, -sqrt(8)i Atoms = cos (sqrt(2)t), sin(sqrt(2)t), cos(sqrt(8)t), sin(sqrt(8)t) Vector x(t) = vector linear combination of the above 4 atomsMaple routines for second orderde1:=diff(x(t),t,t)=-6*x(t)+4*y(t); de2:=diff(y(t),t,t)=2*x(t)-4*y(t); dsolve({de1,de2},{x(t),y(t)}); x(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)+_C3*sin(2*sqrt(2)*t)+_C4*cos(2*sqrt(2)*t), y(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)-(1/2)*_C3*sin(2*sqrt(2)*t)-(1/2)*_C4*cos(2*sqrt(2)*t)}Eigenanalysis method section 7.4u(t) = (a1 cos(sqrt(2)t) + b1 sin(sqrt(2)t)) v1 + (a2 cos(sqrt(8)t) + b2 sin(sqrt(8)t)) v2 where (-2,v1), (-8,v2) are the eigenpairs of A. The two vector terms in u(t) are called the natural modes of oscillation. The natural frequencies are sqrt(2), sqrt(8). Eigenanalysis of A gives v1=[1,1], v2=[2,-1].Railway cars. Problem 7.4-24Cayley-Hamilton-Ziebur method Laplace Resolvent method for second order Eigenanalysis method section 7.4

Non-Homogeneous SystemsDirect solution methods with the Laplace Resolvent Computer Algebra System methods Variation of Parameters Formula for systems Exercise solutions: ch7 and ch8.Extra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.Extra Credit Maple Project: Earthquakes. Explore a 5-story or 7-story building and the resonant frequencies of oscillation of the building which might make it destruct during an earthquake. See Edwards-Penney, application section in 7.4.

Dynamical Systems TopicsEquilibria. Stability. Instability. Asymptotic stability. Classification of equilibria for u'=Au when det(A) is not zero, for the 2x2 case.

Spiral, saddle, center, node.Linearization theory. Jacobian.Detecting stability: Re(lambda)<0 ==> asym. stability. Stability at t=-infinity classifiesUnstablesolutions. Maple phase diagram tools. Demonstration for the example x' = x + y, y' = 1 - x^2 How to detect saddle, spiral, node, center in the linear case using Zeibur's method and examples. Limitations: In the case of a node, we cannot sub-classify as improper or proper using the Zeibur method and examples. The finer sub-classifications require the exponential matrix e^{At} or else a synthetic eigenvalue theorem which calculated the sub-classification.

B>Nonlinear stability theory When the linearized classification and stability transfers to the nonlinear system. stability of almost linear [nonlinear] systems, phase diagrams, classification of nonlinear systems.Nonlinear stabilityphase diagrams, classification. Using DEtools and DEplot in maple to make phase diagrams. Jacobian.