The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 158 Wednesday, February 4, 2004 12:49 PM

158 The Mathematics of Financial Modeling and Investment Management

tr(AB+ ) = trA + trB

The operation of addition of vectors defined above is clearly a special

case of the more general operation of addition of matrices.

Multiplication

Consider a scalar c and a matrix:

A = {}aijnm

The product cA = Ac is the n×m matrix obtained by multiplying each

element of the matrix by c:

cA = Ac = { }caij nm

Multiplication of a matrix by a scalar is associative with respect to

matrix addition:

c(AB+ )= cA + cB

Let’s now define the product of two matrices. Consider two matrices:

A = {}aitnp

and

B = { }bsj pm

The product C = AB is defined as follows:

 p 

C = AB = {}cij = ∑aitbtj

t = 1 

The product C = AB is therefore a matrix whose generic element {cij} is

the scalar product of the ith row of the matrix A and the jth column of

the matrix B. This definition generalizes the definition of scalar product

of vectors: The scalar product of two n-dimensional vectors is the product

of an n×1 matrix (a row vector) for a 1×n matrix (the column vector).

Following the above definition, the matrix product operation is per-

formed rows by columns. Therefore, two matrices can be multiplied