The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 143 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 143

a =

a 1

a 2

·

·

aN

where an is the exposure of asset n to attribute a.

Vector components can be either real or complex numbers. Return-

ing to the row vector w of a portfolio of holdings, a positive value for

wn would mean that some of the risky asset n is held in the portfolio; a

value of zero would mean that the risky asset n is not held in the portfo-

lio. If the value of wn is negative, this means that there is a short posi-

tion in risky asset n.

While in most applications in economics and finance vector compo-

nents are real numbers, recall that a complex number is a number which

can be represented in the form

c = a + bi

where i is the imaginary unit. One can operate on complex numbers^2 as if

they were real numbers but with the additional rule: i^2 = –1. In the follow-

ing we will assume that vectors have real components unless we explicitly

state the contrary.

Vectors admit a simple graphic representation. Consider an n-dimensional

Cartesian space. An n-dimensional vector is represented by a segment

that starts from the origin and such that its projections on the n-th axis

are equal to the n-th component of the vector. The direction of the vec-

tor is assumed to be from the origin to the tip of the segment. Exhibit

5.1 illustrates this representation in the case of the usual three spatial

dimensions x,y,z.

The (Euclidean) length of a vector x, also called the norm of a vec-

tor, denoted as x , is defined as the square root of the sum of the

squares of its components:

x = x^22

1 + ... + xn

(^2) In rigorous mathematical terms, complex numbers are defined as ordered pairs of
real numbers. Operations on complex numbers are defined as operations on pairs of
real numbers. The representation with the imaginary unit is a shorthand based on a
rigorous definition of complex numbers.