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Wednesday January 14
−
Lecture 5 :
Matrix algebra II.
(Refers to section 3.1)
Expectations:
1.
Define matrix multiplication
2.
Perform block multiplication.
5.0
Example
−
If
A
is a 3
×
3 matrix with rows vectors
r
A
1
,
r
A
2
,
r
A
3
n
. and
B
is an 3
×
3
matrix with column vectors
c
B
1
,
c
B
2
,
c
B
3
.
Find an expression which represents the (
i
,
j
)
th
entry of
B
T
A
T
in terms of the rows and columns of
A
and
B
. If the entries of
A
are 1, 2, 3,
…., 9 (listed horizontally) and the entries of
B
are 10, 11, 12,… 18, find the (2, 3)th entry
of
B
T
A
T
.
Solution:
•
Known: the rows of
B
T
are the columns of
B
. So
B
T
has rows
c
B
1
,
c
B
2
,
c
B
3
.
•
Known: the columns of
A
T
are the rows of
A
. So
A
T
has columns
r
A
1
,
r
A
2
,
r
A
3
.
•
Let
D
= [
d
ij
] =
B
T
A
T
.
•
Then by definition of the product of matrices
d
ij
= <row
i
of
B
T
, column
j
of
A
T
> .
= <
c
Bi
,
r
Aj
.> .
•
We seek the entry
d
23
.
We have
d
23
= <
c
B
2
,
r
A
3
.>
= < (11 , 14, 17) , (7 , 8, 9)> =
77 + 112 + 153 = 342.
5.1
Remark
−
We now show that the dot-product of two vectors is a number that can be
expressed as a product of two matrices.
o

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