Encyclopedia of Financial Models, 3 Volume Set

Fundamentals of Matrix Algebra

SERGIO M. FOCARDI, PhD

Partner, The Intertek Group

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: Ordinary algebra deals with operations such as addition and multiplication performed on individual numbers. In many applications, however, it is useful to consider operations performed on ordered arrays of numbers. This is the domain of matrix algebra. Ordered arrays of numbers are called vectors and matrices while individual numbers are called scalars.

In financial modeling, it is useful to consider operations performed on ordered arrays of numbers. Ordered arrays of numbers are called vectors and matrices while individual numbers are called scalars. In this entry, we will discuss some concepts, operations, and results of matrix algebra used in financial modeling.

VECTORS AND MATRICES DEFINED

We begin by defining the concepts of vector and matrix. Though vectors can be thought of as particular matrices, in many cases it is useful to keep the two concepts—vectors and matrices—distinct. In particular, a number of important concepts and properties can be defined for vectors but do not generalize easily to matrices.1

Vectors

An n-dimensional vector is an ordered array of n numbers. Vectors are generally indicated with boldface lowercase letters, although we do not always follow that convention in this book. Thus a vector x is an array of the form:

The numbers ai are called the components of the vector x.

A vector ...