# 2250-4 12:55pm Lecture Record Week 13 S2010

Last Modified: April 15, 2010, 07:35 MDT.    Today: August 16, 2018, 00:43 MDT.

## 13 Apr: Sections 6.1, 6.2, 7.3

```Circuits EPbvp3.7:
Electrical resonance.
Derivation from mechanical problems 5.6.
THEOREM: omega = 1/sqrt(LC).
Impedance, reactance.
amplitude
Transfer function.
Input and output equation.
EIGENANALYSIS WARNING
Reading Edwards-Penney Chapter 6 may deliver the wrong ideas
about how 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.
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 called Fourier'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 Model

THEOREM. 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.

Drill Problems

In 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 independent

1. 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.
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.

Determinant problem from chapter 3: B(n+1)=2B(n)-B(n-1). This
is a second order difference equation.

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).
Survey of Methods for solving a 2x2 dynamical system
1. 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). 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
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.
```

## 16 Apr: Second Order Systems. Section 7.4

```Second Order Systems

How 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.

The model u'' = Ax + F(t)
Coupled Spring-Mass System. Problem 7.4-6
A:=matrix([[-6,4],[2,-4]]);
Railway cars. Problem 7.4-24
Earthquake model
Cayley-Hamilton-Ziebur method
Laplace Resolvent method for second order
Maple routines for second order

```

## 16 Apr: Non-Homogeneous Systems. Section 8.2

```Non-Homogeneous Systems
Direct solution methods with the Laplace Resolvent
Computer Algebra System methods
Variation of Parameters Formula for systems

Exercise solutions: ch7 and ch8.
```
Systems of Differential Equations references
Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)

Extra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.