1
COMPUTER SCIENCE 349A
Handout Number 17
MATRIX INVERSES
This topic is discussed in the textbook in Section 10.2 in terms of an
LU
decomposition (which is just another way of interpreting Gaussian elimination).
We are
omitting all of Chapter 10.
The following material is similar to that in Section 10.2 but is
not described in terms of an
LU
decomposition.
If it is necessary to compute
A
−
1
, this is most efficiently and accurately done
using Gaussian elimination (with partial pivoting)
and the fact that
I
A
A
=
−
1
.
Suppose that a matrix
A
is given.
If the (unknown) column vectors of
A
−
1
are
denoted
)
(
)
2
(
)
1
(
,
,
,
n
x
x
x
K
then
[
]
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
×
=
−
1
0
0
0
1
0
0
0
1



)
(
)
2
(
)
1
(
1
L
M
O
M
L
L
L
n
x
x
x
A
A
A
,
and the vectors
x
(
i
)
can be determined by solving the
n
linear systems
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
1
0
0
,
,
0
1
0
,
0
0
1
)
(
)
2
(
)
1
(
M
L
M
M
n
x
A
x
A
x
A
.
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 Spring '11
 Oadje
 Computer Science, Gaussian Elimination, floatingpoint operation, 1 0 L, 0 1 l, 0 0 L

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