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Appendix

d

Fundamentals of Matrix Algebra

i

n financial econometrics, 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 concepts, operations, and results of matrix algebra.

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

Vectors

An n-dimensional vector is an ordered array of n numbers. Vectors are gener-
ally indicated with boldface lowercase letters, although we do not always fol-
low that convention in the textbook. Thus a vector x is an array of the form:

x=[]xx 1 ,..., n

The numbers ai 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]

(^1) Vectors can be thought as the elements of an abstract linear space while matrices
are operators that operate on linear spaces.