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Matrix inversion
Inverting Matrix
±
AX = I
→
X = A
1
±
How hard is the matrix inversion problem?
Theorem: Multiplication is as hard as inversion
Proof: Let I(n) be the cost of inversion. Let M(n) be the
cost of multiplication. Want to show I(n) =
Θ
(M(n)).
±
M(n)=O(I(n))
√
±
I(n) = O(M(n))
Inverting Matrix
±
AX = I
→
X = A
1
±
How hard is the matrix inversion problem?
Theorem: Multiplication is as hard as inversion
Proof: Let I(n) be the cost of inversion. Let M(n) be the cost
of multiplication. Want to show I(n) =
Θ
(M(n)).
±
M(n)=O(I(n))
±
I(n) = O(M(n))
example
±
Two other problems:
±
Multiplication of two ndigit numbers x and y
±
Squaring one ndigit number z
±
Which problem is harder (assuming +/ can be done
easily)?
xy = ((x+y)
2
(xy)
2
)/4
Inverting Matrix
±
M(n)=O(I(n))
Consider A, B, C s.t. AB = C
Construct
±
I(n) = O(M(n))
I
DD
I
B
I
AB
A
I
D
I
B
I
A
I
D
n
n
n
n
n
n
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
−
−
1
1
0
0
0
0
0
0
0
,
,
.
3
assumming
)),
(
(
)
(
)
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 Spring '09

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