The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 156 Wednesday, February 4, 2004 12:49 PM

156 The Mathematics of Financial Modeling and Investment Management

As another example, a portfolio’s excess return is found by taking

the transpose of the excess return vector, r, and multiplying it by the

vector of portfolio weights, w. That is,

r

T

⋅ w

r 1

r 2

·

·

rN

= w 1 w 2 ......wN

N

= ∑rnwN

n = 1

Two vectors x, y are said to be orthogonal if their scalar product is

zero. The scalar product of two vectors can be interpreted geometrically

as an orthogonal projection. In fact, the inner product of vectors x and

y, divided by the square norm of y, can be interpreted as the orthogonal

projection of x onto y. The following two properties are an immediate

consequence of the definitions:

x = x ⋅x

( ax)⋅( by) = abxy⋅

Matrix Operations

The following five operations on matrices are usually defined: (1) trans-

pose, (2) addition, (3) multiplication, (4) inverse, and (5) adjoint.

Transpose

The definition of the transpose of a matrix is an extension of the trans-

pose of a vector. The transpose operation consists in exchanging rows

with columns. Consider the n×m matrix

A = {}aijnm

The transpose of A, denoted AT or A′is the m×n matrix whose ith row is

the ith column of A:

A

T

= {}aji mn