The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 157 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 157

The following should be clear from this definition:

(AT)

T

=A

and that a matrix is symmetric if and only if

A

T

=A

Addition

Consider two n×m matrices

A={}aijnm

and

B={}bij nm

The sumof the matrices Aand Bis defined as the n×m matrix obtained

by adding the respective elements:

AB+ ={aij +bij}nm

Note that it is essential for the definition of addition that the two matri-

ces have the same order n×m.

The operation of addition can be extended to any number N of

summands as follows:

N



N



∑Ai =∑asij





s = 1 s = 1 nm

where asij is the generic i,j element of the sth summand.

The following properties of addition are immediate from the defini-

tion of addition:

AB+ BA+ =

A BC +( + ) =(AB+ )+C=ABC++