The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 155 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 155

λx = [λx 1 ...λxn ]

As an example of the multiplication of a vector by a scalar, consider

the vector of portfolio weights w = [w 1 ...wn]. If the total portfolio value

at a given moment is P, then the holding in each asset is the product of

the value by the vector of weights:

Pw = [Pw 1 ...Pwn ]

A similar definition holds for column vectors. It is clear from this defini-

tion that

ax = a x

and that multiplication by a scalar is associative as

a(xy+ ) = ax + ay

The scalar (or inner) product of two vectors of the same dimensions

x, y, denoted as x · y, is defined between a row vector and a column vec-

tor. The scalar product between two vectors produces a scalar according

to the following rule:

n

xy⋅ = (^) ∑ xiyi
i = 1
For example, consider the column vector a of a particular attribute dis-
cussed earlier and the row vector w of portfolio weights. Then a · w is a
scalar that shows the exposure of the portfolio to the particular
attribute. That is,
aw⋅
a 1
a 2
·
·
aN
= w 1 w 2 ...... wN
N

= ∑ anwN

n = 1