The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 154 Wednesday, February 4, 2004 12:49 PM

154 The Mathematics of Financial Modeling and Investment Management

x 1

T ·

x = (^) ·
·
xn
Clearly the transpose of the transpose is the original vector:
(x T)
T
= x
Addition
Two row (or 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+ = [x 1 + y 1 ...xn + yn]
This definition can be generalized to any number N of summands:
N N N

∑ xi = ∑ x 1 i...∑ yni

i = 1 i = 1 i = 1

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 commu-

tative 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.

Multiplication

We define two types of multiplication: (1) multiplication of a scalar and

a vector and (2) scalar multiplication of two vectors (inner product).^4

The multiplication of a scalar λ and a row (or column) vector x,

denoted as λx, is defined as the multiplication of each component of the

vector by the scalar:

(^4) Different types of products between vectors can be defined: the vector product be-
tween vectors produces a third vector and the outer product produces a matrix. We
do not define them here, as, though widely used in the physical sciences, they are not
typically used in economics.