LECTURE 6
§
LU factorization
§
LDU factorization
§
Uniqueness
§
LU factorization of non-square matrices
A little detour into Chapter 2 that we should
know
@2019 Arizona State University.

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
Consider system of equations
x
+
2
y
+
z
=
2
3
x
+
8
y
+
z
=
12
4
y
+
z
=
2
Put it in a matrix form
x
y
z
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
2
12
2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
Start row eliminations of A
§
How can we describe this operation with the help
of a matrix multiplication?
§
I.e. what would be the matrix E
21
, such that E
21
A
gives us the matrix on the right?

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
Start row eliminations of A
E
21
A

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
Start row eliminations of A
§
OK, we are finished with the first pivot: it has only zeroes
beneath it.
§
We got lucky: we only had to perform E
21
operation (worked
with 2
nd
row, 1
st
column)
§
We did not have to perform E
31
operation, because there is
already zero there, otherwise we would!

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
In this case, for consistency, we can say that E
31
is the identity matrix
E
31
=

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
OK, now the second pivot
§
Since it is a 3x3 matrix, after working with the
second pivot, we already got
U
!
§
What is E
32
here?

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
OK, now the second pivot
E
32
E
21
A=
E
21
A
E
32
U

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
E
32
E
21
A=
E
21
A
E
32
U
E
ij
is called an elementary matrix that subtracts
l
times row
j
from row
i
. It is the identity matrix with a
–
l
in row
i
, column
j
.

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
Therefore, elimination in a matrix form is
E
32
E
31
E
21
A=U
§
Where E
ij
before A are all the row operations we
had to perform.
§
For bigger matrices, of course, there are many
more operations and many more contributing E
ij
!

@2019 Arizona State University.
LU FACTORIZATION: ROW OPERATIONS
How do we inverse elimination step, i.e. what is E’
ij
such that
E’
ij
E
ij
A=A, or in other words E’
ij
E
ij
=I?

#### You've reached the end of your free preview.

Want to read all 42 pages?

- Fall '19
- Yulia Peet