1
5. The LU factorization
5.1.1 The factorization
If the
1

n
leading principal minors of
n
n
×
ℜ
∈
A
are all nonsingular then there
exist
L
and
U
such that
LU
A
=
,
where
L
is a lower triangular matrix with unit diagonal elements
=
1
1
1
2
1
21
L
O
M
M
n
n
l
l
l
0
L
and
U
is an upper triangular matrix
=
nn
n
n
u
u
u
u
u
u
0
U
M
O
L
L
2
22
1
12
11
.
5.1.2 Algorithm for Factorization
We will demonstrate the factorization algorithm for the case
3
=
n
. The extension
to higher dimensions is obvious.
We first write the factorization in component form
=
33
32
31
23
22
21
13
12
11
33
23
22
13
12
11
32
31
21
1
1
1
a
a
a
a
a
a
a
a
a
u
u
u
u
u
u
l
l
l
.
Form the first column of
A
we obtain
11
11
a
u
=
11
21
21
21
11
21
u
a
l
a
u
l
=
⇒
=
11
31
31
31
11
31
u
a
l
a
u
l
=
⇒
=
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