Anon

394 The Basics of financial economeTrics

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

AT={}ajimn

The following should be clear from this definition:

()AAT =

T

and that a matrix is symmetric if and only if

AAT=

Addition Consider two n × m matrices:

A={}aij nm and B={}bijnm

The sum of the matrices A and B is defined as the n × m matrix obtained by
adding the respective elements:

AB+={}abij+ ij nm

Note that it is essential for the definition of addition that the two matrices
have the same order n × m.
The operation of addition can be extended to any number N of sum-
mands as follows:

Ai

s

N

s

s

N

nm

aij

==

∑∑=















(^11) 
where asij is the generic i,j element of the sth summand.
Multiplication Consider a scalar c and a matrix
A={}aijnm
The product cA = Ac is the n × m matrix obtained by multiplying each ele-
ment of the matrix by c:
ccAA=={}caij nm