Wednesday March 11
−
Lecture 26 :
Matrix factorization
(Refers to 3.5)
Expectations
:
1.
Define the diagonal, upper triangular, lower triangular and triangular for an
m
×
n
matrix
A
.
2.
Define the
LU
factorization of an
m
×
n
matrix
A
.
26.1
Definition
−
If
A
is an
m
×
n
matrix [
a
ij
]
m
×
n
we will call the
diagonal of A
those
entries
a
11
,
a
22
,
a
33
, .
..,
a
mm
. Notice that we are defining the diagonal of a matrix which is
not
necessarily square.
•
An
m
×
n
matrix
A
where all entries above the main diagonal are zeroes is called a
lower triangular
m
×
n
matrix. That is if
j
>
i
, then
a
ij
= 0.
•
An
upper triangular
m
×
n
matrix is one where all entries below the main
diagonal are zeroes.
That is if
i
>
j
, then
a
ij
= 0.
•
A
triangular
m
×
n
matrix
A
is one which is either upper or lower triangular.
26.1.1
Examples of upper triangular matrices:
⎡
1
−
1
03
⎤
⎡
02105
⎤
⎡
111
⎤
⏐
0 211
⏐
⏐
00031
⏐
⏐
0
−
1
1
⏐
⎣
00
−
3
0
⎦
⎣
00101
⎦
⎣
000
⎦
26.1.2
Remark
−
Triangular matrices are of
interest since systems
A
x
=
b
where the
coefficient matrix
A
is triangular are easy to solve. An example is given.
x
1
+
4
x
2
−
3
x
3
−
x
4
+
5
x
5
=
4
5
x
3
+
x
4
+
x
5
=
8
2
x
5
= 6
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View Full DocumentThis system is easily seen to have the solution
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
= (
−
9
−
2
x
2
+ (2/5)
x
4
,
x
2
, 1
−
(1/5)
x
4
,
x
4
,
3)
26.2
Lemma (
Optional
)
−
The product of two upper triangular matrices
is again upper
triangular. The product of two lower triangular matrices is again lower triangular.
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 Winter '08
 All
 ax, Ly, lower triangular

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