CHARACTERISTIC ROOTS AND VECTORS
1. DEFINITION OF CHARACTERISTIC ROOTS AND VECTORS 1.1. Statement of the characteristic root problem. Find values of a scalar for which there exist vectors x = 0 satisfying Ax = x (1)
where A is a given nth order matrix. Th
REVIEW OF SIMPLE UNIVARIATE CALCULUS
1. APPROXIMATING
CURVES WITH LINES
1.1. The equation for a line. A linear function of a real variable x is given by y = f(x) = ax + b, a and b are constants (1)
The graph of linear equation is a straight line. The numb
INTRODUCTION TO MATRIX ALGEBRA
1. DEFINITION
OF A MATRIX AND A VECTOR
1.1. Denition of a matrix. A matrix is a rectangular array of numbers arranged into rows and columns. It is written as a11 a12 . a1n a21 a22 . a2n . . (1) . . . . am1 am2 . amn The abov
FUNCTIONS AND EQUATIONS
1. SETS
AND SUBSETS
1.1. Denition of a set. A set is any collection of objects which are called its elements. If x is an element of the set S, we say that x belongs to S and write x S. If y does not belong to S, we write y S. The s
GEOMETRY OF MATRICES
1. SPACES
n n
OF
VECTORS
1.1. Denition of R . The space R consists of all column vectors with n components. The components are real numbers. 1.2. Representation of Vectors in Rn. 1.2.1. R2. The space R2 is represented by the usual x1
SINGLE VARIABLE OPTIMIZATION
1. DEFINITION OF LOCAL MAXIMA AND LOCAL MINIMA 1.1. Note on open and closed intervals. 1.1.1. Open interval. If a and b are two numbers with a < b, then the open interval from a to b is the collection of all numbers which are
SIMPLE CONSTRAINED OPTIMIZATION
1. INTUITIVE INTRODUCTION
TO
CONSTRAINED OPTIMIZATION
Consider the following function which has a maximum at the origin. y = f (x1 , x2 ) = 49 x2 x2 1 2 The graph is contained in gure 1. (1)
FIGURE 1. The function y = 49 x2
SIMPLE MULTIVARIATE CALCULUS
1. REAL-VALUED FUNCTIONS OF SEVERAL VARIABLES 1.1. Denition of a real-valued function of several variables. Suppose D is a set of n-tuples of real numbers (x1 , x2 , x3 , . . . , xn ). A real-valued function f on D is a rule t
SIMPLE MULTIVARIATE OPTIMIZATION
1. DEFINITION
OF LOCAL MAXIMA AND LOCAL MINIMA
2
1.1. Functions of 2 variables. Let f(x1 , x2 ) be dened on a region D in
containing the point (a, b). Then
a: f(a, b) is a local maximum value of f if f(a, b) f(x1 , x2 ) fo
SYSTEMS OF EQUATIONS
1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which we would like to solve. The variables are typical
CONVEXITY AND OPTIMIZATION
1. CONVEX
SETS
1.1. Denition of a convex set. A set S in Rn is said to be convex if for each x1 , x2 S, the line segment x1 + (1-)x2 for (0,1) belongs to S. This says that all points on a line connecting two points in the set ar
THE IMPLICIT FUNCTION THEOREM
1. A
SIMPLE VERSION OF THE IMPLICIT FUNCTION THEOREM
1.1. Statement of the theorem. Theorem 1 (Simple Implicit Function Theorem). Suppose that is a real-valued functions dened on a domain D and continuously differentiable on