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FACTORING MATRICES
TERRY A. LORING
A lot of factorization results can be done using row operatations paired with
column operations.
To illustratate, suppose we have factored a
3
by3 matrix
A
as
A
=
ST,
for some
3
by3 matrices. We sort of like this factorization, but want a nicer
one. We can insert
I
=
1
0
1
0
1
0
0
0
1

1
1
0
1
0
1
0
0
0
1
=
1
0

1
0
1
0
0
0
1
1
0
1
0
1
0
0
0
1
in the middle to get
A
=
S
1
0

1
0
1
0
0
0
1
1
0
1
0
1
0
0
0
1
T
.
If we let
S
1
=
S
1
0

1
0
1
0
0
0
1
and
T
1
=
1
0
1
0
1
0
0
0
1
T
we have
A
=
S
1
T
1
which might be better than the factorization we had before.
This is a lot of writing of zeros.
We know about how elementary matrices work when multiplied on the left
of other matrices. They implement row operations. Multiplied on the
right
they implement
column operations.
If
S
=
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
then
S
1
=
x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
1
0

1
0
1
0
0
0
1
=
x
1
y
1
z
1

x
1
x
2
y
2
z
2

x
2
x
3
y
3
z
3

x
3
.
1
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TERRY A. LORING
On the other hand, if
T
=
a
1
a
2
a
3
b
1
b
2
b
3
c
1
c
2
c
3
then
T
1
=
1
0
1
0
1
0
0
0
1
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
=
a
1
+
a
3
b
1
+
b
3
c
1
+
c
3
a
2
b
2
c
2
a
3
b
3
c
3
.
That is,
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 Spring '10
 Coluos
 Factoring, Matrices

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