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1
Suppose
A
3
is to be computed for the 4
×
4 matrix
A
below, which
is “partitioned” as a 2
×
2 “block matrix”.
Suppose the “block matrix” is treated as an ordinary matrix
containing scalar entries and its power computed accordingly:
It can be verified by direct computation of
A
3
that the above result
is also correct.
Matrix partition and block matrices:
[]
,
ij
ij
AA
BB
⎡
⎤
⎡⎤
⎢
⎥
==
⎢⎥
⎢
⎥
⎣⎦
⎢
⎥
⎣
⎦
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Compatible partitions for block matrix multiplication:
column partition of the first matrix = row partition of the
second matrix
Block matrix multiplication with
compatible
partitions
[]
ik
ij
jk
j
CA
BC
A
B
⎡
⎤
⎡⎤
==
=
=
⎢
⎥
⎣⎦
⎣
⎦
∑
Example
11
12
21
22
A
A
A
A
=
⎢⎥
11
12
B
B
B
B
⎡
⎤
=
⎢
⎥
⎣
⎦
A
11
B
11
+
A
12
B
21
:
A
11
B
12
+
A
12
B
22
:
A
21
B
11
+
A
22
B
21
:
A
21
B
12
+
A
22
B
22
:
3
12
1
2
1
2
1
2
Then
.
p
m
nn
n
n
AB
A
A
A
B
B
B
⎡⎤
=
⎣⎦
′
⎢⎥
′
=
′
′
′
′
′′
==
+
+
+
′
←→
↑↑
↑
↓↓
↓
bb
b
a
a
a
b
b
aa
a
a
ba
b
a
b
b
"
#
""
#
[]
11
22
Let
and
.
np
mn
AB
′
′
⎡
⎤⎡
⎤
⎢
⎥⎢
⎥
′
′
⎢
⎥
=
=
⎢
⎥
⎢
⎥
′
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 Spring '09
 Fong

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