Financial Economics

WEB-APPENDIX O^*

FUNDAMENTALS OF MATRIX ALGEBRA

In many applications in financial economics, 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 appendix, we will discuss the some concepts, operations, and results of matrix algebra used in this textbook.

O.1 VECTORS AND MATRICES DEFINED

Let's now define precisely 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

O.1.1 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 the textbook. Thus, a vector x is an array of the form:

x = [x₁,…, x_n]

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

A vector is identified by the set of its components. Vectors can be row vectors or column vectors. If the vector components appear in a horizontal row, then the vector is called a row vector, as for instance the vector:

x = [1,2,8,7]

Here are two examples. Suppose that we let w_n be a risky asset's weight in a portfolio. Assume that there are N risky assets. Then the following vector, w, is a row vector that represents a portfolio's ...