Matrix Arithmetic
If you don’t remember how to add matrices, you should look it up now. Here we are
going to talk about matrix products.
Let
A
∈
R
m
×
n
and
B
∈
R
n
×
p
. Let’s also say that the matrix
A
is the coordinate
representation of a linear transformation
A
:
R
n
→
R
m
, and likewise for
B
and
B
:
R
p
→
R
n
. Then linear transformation
C
=
AB
:
R
p
→
R
n
→
R
m
has as its coordinate representation the matrix
C
=
AB
. While it is true that
a
matrix is a rectangular array of numbers
, it will be useful for us to remember that a
matrix represents a linear function from one vector space to another:
a matrix is a
linear transformation
. This is precisely why the natural product of two matrices
isn’t entrywise, like addition, but instead has the (not as intuitive) form
C
= [
c
ij
]
,
where
c
ij
=
n
X
k
=1
a
ik
b
kj
.
Let’s let
a
t
i
be the
i
th
row of
A
, and
b
j
be the
j
th
column of
B
. Then
c
ij
=
a
t
i
b
j
. If
we now let
c
j
be the
j
th
column of
C
, we can write
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.
 Fall '10
 MARK

Click to edit the document details