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anda3
3 lower triangular matrix appears as
⎡
⎣
⎤
⎦
L
11
00
L
=
L
21
L
22
0
L
31
L
32
L
33
Triangular matrices play an important role in linear algebra,since they simplify
many computations.For example, consider the equations
Lx
=
c
,or
L
11
x
1
=
c
1
L
21
x
1
+
L
22
x
2
=
c
2
L
31
x
1
+
L
32
x
2
+
L
33
x
3
=
c
3
.
If we solve the equationsforward,starting with the first equation, the computations
are very easy,since each equationwould contain only one unknown at a time. The
solutionwould thus proceedas follows:
x
1
=
c
1
/
L
11
x
2
=
(
c
2
−
L
21
x
1
)
/
L
22
x
3
=
(
c
3
−
L
31
x
1
−
L
32
x
2
)
/
L
33
.
This procedure isknownas
forwardsubstitution
. Inasimilarway,
Ux
encountered
in Gauss elimination,can easilybe solvedby
back substitution
, which starts with the
last equation and proceeds backward through the equations.
The equations
LUx
=
c
,
b
, which are associatedwith LU decomposition,can also
be solved quicklyif we replace themwith two sets of equivalentequations:
Ly
=
=
b
and
Ux
=
y
. Now
Ly
=
b
can be solved for
y
by forward substitution, followedbythe
solution of
Ux
y
by meansofback substitution.
The equations
Ix
=
=
c
, which are producedbyGauss-Jordan elimination, are
equivalentto
x
=
c
(recall the identity
Ix
=
x
), so that
c
is already the solution.
EXAMPLE 2.1
Determine whether the following matrix issingular:
⎡
⎣
⎤
⎦
2
.
1
−
0
.
61
.
1
A
=
3
.
2
4
.
7
−
0
.
8
3
.
1
−
6
.
5
4
.
1
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