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++