An Example on Versions of LP Formulation
Problem source: Operations Research, Applications and Algorithms by Winston (Third
Edition, older version), page 92 problem 11
Eli Daisy produces the drug Rozac from four chemicals. Today they must produce
1000 lb

DC Motor Position: State-Space Methods for Controller
Design
Key MATLAB commands used in this tutorial are: ss , order , det , ctrb , place , step
Contents
Designing the full state-feedback controller
Disturbance response
Adding integral action
From the m

Application of Determinant to Systems:
Cramer's Rule
We have seen that determinant may be useful in finding the inverse of a nonsingular matrix. We
can use these findings in solving linear systems for which the matrix coefficient is nonsingular
(or invert

Characteristic polynomial
This is the determinant of
[1 1
0 ]
[ 0 2 0 ]
[ 0 1 2]
which is (expand along the first column) (1-)(2-)2.
Eigenvalues
We can read off the roots. They are 1 and 2 (the root 2 has multiplicity 2). These are the
eigenvalues.
Bases

Chapter 3
Classication of PDEs and Related
Properties
3.1
Linear Second Order PDEs in two Independent Variables
The most general form of a linear, second order PDE in two independent variables x, y and the
dependent variable u(x, y) is
2u
u
2u
u
2u
+C

The Case of Complex Eigenvalues
First let us convince ourselves that there exist matrices with complex eigenvalues.
Example. Consider the matrix
The characteristic equation is given by
This quadratic equation has complex roots given by
Therefore the matri

FINDING EIGENVALUES AND EIGENVECTORS
EXAMPLE 1: Find the eigenvalues and eigenvectors of the matrix
1 3 3
A = 3 5 3 .
6 6 4
SOLUTION:
In such problems, we rst nd the eigenvalues of the matrix.
FINDING EIGENVALUES
To do this, we nd the values of which sa

37
Homogeneous Systems with Constant Coecients: Repeated Eigenvalues
In this section we consider the case when the characteristic equation possesses
repeated roots. A major diculty with repeated eigenvalues is that in some
situations there is not enough l

Find the roots of a polynomial using its companion matrix
I would like to find the roots of a polynomial using its companion matrix.
The polynomial is p(x)=x 4 10x 2 +9
The companion matrix M is
A theorem says that the eigenvalues of M are the roots of p(

Go to Tools on top of Excel sheet. Click on the Solver. You will see solver tool
dialogue box shown below.
Enter objective function
cell
Select max or min
Enter decision variables
cells, use comma to enter
non-continuous cells
Enter constraints by
clickin

SYSEN 530
Problem Set No. 1
1.
5
1
Let a = 4 and b = 1 . What are the components of the vector x whose
0
1
end point falls halfway between a and b on the straight line connecting them?
2. The closed half space
2 x1 3 x 2 6 x3 x 4 2
1
1
cont

Page 75 problem 1
In the post office example, suppose that each full-time employee works eight hours per day.
Thus, Monday's requirement of 17 workers may be viewed as a requirement of 8x17 =136
hours. The post office may meet its daily labor requirements

PROBLEM DESCRIPTION:
State of the Ott Electronics is starting to manufacture their newest product, the FrappZapper. The Frapp-Zapper consists of four modules, any number of which can be
called out in the purchasers order. There are five orders in the firs

Winston Chapter 3.4, Page 73, Number 1 (Linear Programming).
1
Winston Chapter 3.4, Page 73, Number 1 (Linear Programming).
Problem Statement: There are three factories on the Momiss River (1, 2, and 3). Each
emits two types of pollutants (1 and 2) into t

The Planning committee of a bank makes monthly decisions on the amount of funds to
allocate to loans and to government securities. Some of the loans are secured (backed by
collateral such as a home or automobile), and some are unsecured. A list of the var

Lotka-Volterra Equation Analysis
Aim
To have a basic understanding of the Lotka-Volterra (LV) Equation on how it resulted in a
unstable oscillation
To study to the key parameters for their effects on the amplitude and the frequency of output
oscillation
F

_+_*_;+;u_
PROBLEMS.
and wheat to plant this year. An acre of wheat yields 25
bushels of wheat and requires 10 hours of labor per week.
An acre of corn yields 10 bushels of corn and requires 4
hours of labor per week. All wheat can be sold at $4 a
bushel,

.<-
il
we know that LP is unbounded. This follows because each time we move in the direction
d an amount that increases x3 by one unit, we increase 2 by 9, and we can move as far as
we want in the direction d.
FHBBLEMS
Group A
1 Show that the following

Eigenvalues and Eigenvectors
Definition. Let
(I is the
. The characteristic polynomial of A is
identity matrix.)
A root of the characteristic polynomial is called an eigenvalue (or a characteristic value) of A.
While the entries of A come from the field F