Anon

392 The Basics of financial economeTrics

vectors, and matrices to produce new scalars, vectors, or matrices. The
notion of operations performed on a set of objects to produce another
object of the same set is the key concept of algebra. Let’s start with vec-
tor operations.

Vector Operations

The following three operations are usually defined on vectors: transpose,
addition, and multiplication.

Transpose The transpose operation transforms a row vector into a column
vector and vice versa. Given the row vector x=[]xx 1 ,..., n its transpose,
denoted as xT or x', is the column vector:

=

⋅

⋅

⋅





     





     

x

x

xT

n

1

Clearly the transpose of the transpose is the original vector: ()xxT =

T

.

Addition Two-row (or two-column) vectors x = [x 1 ,... , xn], y = [y 1 ,... , yn]
with the same number n of components can be added. The addition of two
vectors is a new vector whose components are the sums of the components:

xy+=[]xy 11 ++,...,xynn

This definition can be generalized to any number N of summands:

∑∑= ∑















== = 

xi xy,...,

i

N

i

i

N

ni

i

N

1

1

11

The summands must be both column or row vectors; it is not possible to add
row vectors to column vectors.
It is clear from the definition of addition that addition is a commuta-
tive operation in the sense that the order of the summands does not matter:
x + y = y + x. Addition is also an associative operation in the sense that
x + (y + z) = (x + y) + z.