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12. Triangular factorization and positive deFnite matrices
12.1. A detour on triangular factorization
The notation
A
[
j,k
]
=
⎡
⎢
⎣
a
jj
···
a
jk
.
.
.
.
.
.
a
kj
a
kk
⎤
⎥
⎦
for
A
∈
C
n
×
n
and
1
≤
j
≤
k
≤
n
(12.1)
will be convenient.
Theorem 12.1.
Ama
tr
ix
A
∈
C
n
×
n
admits a factorization of the form
A
=
LDU ,
(12.2)
where
L
∈
C
n
×
n
is a lower triangular matrix with ones on the diagonal,
U
∈
C
n
×
n
is an upper triangular matrix with ones on the diagonal and
D
∈
C
n
×
n
is an invertible diagonal matrix, if and only if the submatrices
A
[1
,k
]
are invertible for
k
=1
,... ,n.
(12.3)
Moreover, if the conditions in
(12.3)
are met, then there is only one set of
matrices,
L
,
D
and
U
, with the stated properties for which
(12.2)
holds.
Proof.
Suppose Frst that the condition (12.3) is in force.
Then, upon
expressing
A
=
·
A
11
A
12
A
21
A
22
¸
in block form with
A
11
∈
C
p
×
p
,
A
22
∈
C
q
×
q
and
p
+
q
=
n
, we can invoke
the Frst Schur complement formula
A
=
·
I
p
O
A
21
A
−
1
11
I
q
¸·
A
11
O
OA
22
−
A
21
A
−
1
11
A
12
I
p
A
−
1
11
A
12
OI
q
¸
repeatedly to obtain the asserted factorization formula (12.2).
Thus, if
A
11
=
A
[1
,n
−
1]
,th
en
α
n
=
A
22
−
A
21
A
−
1
11
A
12
is a nonzero number and
the exhibited formula states that
A
=
L
n
·
A
[1
,n
−
1]
O
Oα
n
¸
U
n
,
where
L
n
∈
C
n
×
n
is a lower triangular matrix with ones on the diagonal
and
U
n
∈
C
n
×
n
is an upper triangular matrix with ones on the diagonal.
The next step is to apply the same procedure to the (
n
−
1)
×
(
n
−
1) matrix
A
[1
,n
−
1]
. This yields a factorization of the form
A
[1
,n
−
1]
=
e
L
n
−
1
·
A
[1
,n
−
2]
O
n
−
1
¸
e
U
n
−
1
,