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A
=
LDM
t
If
A
is nonsingular and
A
=
LU
, then we can set
D
= diag(
U
) and (since
D
is
nonsinglar)
M
t
≡
D

1
U
is a unit upper triangular matrix and
A
=
LDM
t
.
There is no inherent beneﬁt to this factorization over
LU
, but it can give us a
perspective from which to develop other algorithms. The idea is not to compute
LDM
t
from
LU
, but to derive a method to compute
L
,
D
and
M
directly. To that
end, consider the
k
th
column of
A
=
LDM
t
:
a
≡
Ae
k
=
LDM
t
e
k
≡
Ly.
(1)
Suppose we have already know the ﬁrst
k

1 columns of
L
and consider the blocked
treatment of the unit lower triangular system
a
=
Ly
:
±
L
11
0
L
21
L
22
² ±
y
1
y
2
²
=
±
a
1
a
2
²
,
(2)
where
L
11
is
k
×
k
and
known
, but the last column of
L
21
is something we would like
to compute. Forward substitution gives
y
1
as the solution to
L
11
y
1
=
a
1
. Now we
know the ﬁrst
k
elements of
y
, and (from (1))
y
=
DM
t
e
k
,
(3)
giving
e
t
k
y
=
e
t
k
DM
t
e
k
=
d
kk
e
t
k
M
t
e
k
=
d
kk
(right?).
D

1
y
=
M
t
e
k
with
M
t
unit
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 Fall '10
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