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The entries
u
,
v
,
w
,and
x
of the product
A
×
B
can also be produced by the following
matrix-vector product:
⎡
⎤
⎡
⎤
u
w
v
x
e
g
f
h
⎣
⎦
⎣
⎦
=
D
×
We now wr i t e
D
as a sum of seven matrices as shown in Fig.
6.2
;thatis,
D
=
A
1
+
A
2
+
A
3
+
A
4
+
A
5
+
A
6
+
A
7
Let
P
1
,
P
2
,
...
,
P
7
be the products of the
(
n/
2
)
×
(
n/
2
)
matrices
P
1
=
a
+
d
)
×
(
e
+
h
)
P
5
=
a
+
b
)
×
h
P
2
=(
c
+
d
)
×
e
P
6
=(
−
a
+
c
)
×
(
e
+
f
)
P
3
=
a
×
(
f
−
h
)
P
7
=
b
−
d
)
×
(
g
+
h
)
P
4
=
d
×
(
−
e
+
g
)
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
a
+
d
00
a
+
d
0000
0000
a
+
d
00
a
+
d
0 000
c
+
d
000
0 000
A
1
=
A
2
=
−
(
c
+
d
)
000
⎡
⎤
⎡
⎤
000 0
000 0
00
a
−
dd
00
⎣
⎦
⎣
⎦
−
dd
00
0 000
0 000
A
3
=
A
4
=
−
a
00
a
−
a
⎡
⎤
⎡
⎤
000
(
a
+
b
)
000 0
000
a
+
b
000 0
−
0
0
0
0
⎣
⎦
⎣
⎦
0
0
0
0
A
5
=
A
6
=
0
0
0
0
−
a
+
c
−
a
+
c
0
0
⎡
⎤
d
0000
0000
0000
b
−
d
b
−
0
0
⎣
⎦
A
7
=
Figure 6.2
The decomposition of the 4
×
4matrix
D
as the sum of seven 4
×
4matrices.
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