Math 313 Lecture #8
§
1.6: Partitioned Matrices
Matrix Multiplication Again.
Recall that the (
i, j
) entry of the product
AB
for an
m
×
n
matrix
A
= (
a
ij
) and a
n
×
r
matrix
B
= (
b
ij
) is the scalar product (or
inner
product
) of the
i
th
row of
A
and the
j
th
column of
B
:
n
k
=1
a
ik
b
kj
=
a
(
i,
:)
b
j
.
Inherit in this is the partition of
A
into rows and
B
into columns.
Recall also that each column of
AB
is the product of
A
with the corresponding column
of
B
:
AB
= (
Ab
1
, . . . , Ab
r
)
.
This results from another partition of
A
and
B
into
blocks
or
submatrices
: in this case,
A
is left unpartitioned (i.e., taken as itself), and
B
is partitioned into columns.
Another partition of
A
and
B
gives a representation of
AB
in terms of its rows:
AB
=
a
(1
,
:)
B
a
(2
,
:)
B
.
.
.
a
(
m,
:)
B
.
Example.
Let
A
=
2
2
1
3

1
0
and
B
=
3
1

1

2
0
1
.
Then the columns of
AB
are
Ab
1
=
2
2
1
3

1
0
3

1
0
=
4
10
,
Ab
2
=
2
2
1
3

1
0
1

2
1
=
1
5
,
while the rows of
AB
are
a
(1
,
:)
B
=
2
2
1
3
1

1

2
0
1
=
4
1
,
a
(2
,
:)
B
=
3

1
0
3
1

1

2
0
1
=
10
5
.
Although the partitions of
A
and
B
are different, each
block
multiplication leads to the
same thing.
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Block Multiplication.
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