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Week 13: 23 Nov,  24 Nov,  25 Nov,

2250 Lecture Record Week 13 F2009

Last Modified: November 28, 2009, 06:09 MST.    Today: October 21, 2017, 13:43 MDT.

Week 13, Nov 23, 24, 25: Sections 6.1,6.2,7.1,7.2,7.3

23 Nov: Algebraic Eigenanalysis Sections 6.1,6.2

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 Algorithm
   Lawrence Page's pagerank algorithm, google web page rankings.
   Relationship to the power method, stochastic matrices and eigenanalysis.

24 Nov: Algebraic Eigenanalysis. Systems of Differential Equations. Sections 6.2,7.1,7.2,7.3

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 Eigenanalysis
  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.
     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).

25 Nov: Systems of Differential Equations. Ziebur's Lemma. Sections 7.1,7.2,7.3

Systems of two differential equations
   The 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 Method
  ZIEBUR'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 system
  1. 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)=0

Topics 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 models
   The job-site cable hoist example [delayed]
   Sliding plates example  [delayed]
   Home heating example  [more coming]
    References for Eigenanalysis and Systems of Differential Equations.
    Manuscript: Algebraic eigenanalysis (135.8 K, pdf, 07 Apr 2008)
    Manuscript: What's eigenanalysis 2008 (129.5 K, pdf, 07 Apr 2008)
    Manuscript: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008)
    Manuscript: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007)
    Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (111.4 K, pdf, 30 Nov 2009)
    Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
    Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)
    Manuscript: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)
    Slides: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)
    Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
    Text: History of telecom companies (1.1 K, txt, 05 Oct 2008)
    Systems of Differential Equations references
    Manuscript: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)
    Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
    Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)
    Slides: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)
    Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
    Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)