LU - Factorizations
Matrix Factorization
into
Triangular Matrices
Triangular Matrices
A nn square matrix L=(lij) is called
Lower Triangular if lij = 0 if i<j (i.e.
all entries above the diagonal are
zero).
A nn square matrix U=(uij) is called
Upper Triangular if uij = 0 if i>j (i.e.
all entries below the diagonal are
zero).
l11 0
l
 21 l22
l31 l32



ln1 ln 2
0

0 
l33 


ln 3 
0
0 
0


lnn 
Lower Triangular
Matrix
u11 u12
0 u
22

0
0



 0
0
u13  u1n 
u23  u2 n 
u33  u3n 

   
0  unn 
Upper Triangular
Matrix
LU Factorization
The LU Factorization of a nn square Matrix M = (mij) is a way of expressing the
Matrix M as a product of two square nn matrices L and U where the matrix L is a
lower triangular matrix and The matrix U is an upper triangular matrix. Furthermore
the matrix L has all 1's on the diagonal.
 m11
m
 21
M  LU  m31

 
mn1
m12
m22
m32

mn 2
m13  m1n 
m23  m2 n 
m33  m3n  

   
mn 3  mnn 
1 0
l
 21 1
l31 l32



ln1 ln 2
0
0
1

ln 3
 0
 0
 0

 
 1
u11 u12
0 u
22

0
0



 0
0
u13  u1n 
u23  u2 n 
u33  u3n 

   
0  unn 
LU Factorization Method
The factorization is done by keeping tract of a series of elementary matrices where
multiplication on the left by an elementary matrix corresponds to a row operation.
1. Apply the necessary row operations to get the matrix to be upper triangular.
2. For each row operation multiply on the right by an elementary matrix to get U.
u11 u12
0 u
22

Ek  Ek 1    E1  M  U   0
0



 0
0
u13  u1n 
u23  u2 n 
u33  u3n 

   
0  unn 
3. Multiply on the right by the inverse of all of the elementary matrices to get L.

M  E11  E21   Ek1
1 0
l
 21 1
U  L U  l31 l32



ln1 ln 2

0
0
1

ln 3
 0 u11 u12
 0  0 u22
 0  0
0

   

 1  0
0
u13  u1n 
u23  u2 n 
u33  u3n 

   
0  unn 
Elementary Matrices
The elementary matrices Ek must be of a certain type that correspond to multiplying
a row by a number and adding it to another row.
Consider the row operation of
multiplying row i by c and adding
it to row j (i≠j). This row
operation is abbreviated by
Rj=cRi+Rj. The corresponding
elementary matrix has 1's going
down the diagonal, a c in row j
and column i, and zeros in every
other position.
Rj=cRi+Rj
jth row
1
0




0


0

0  0   0
1      
     

        Ek
0  c   

     
0  0   1
ith column
1. The inverse of a matrix like this just puts a –c in the position where c is.
2. The product of two of these type of matrices with a c1 and c2 in two different
positions results in a matrix with c1 in the position where it was and c2 in the
position it was in and 1's on the diagonal and zeros everywhere else.
3. If all row operations have i<j the matrix will be lower triangular.
LU Factorization Method (Example)
The following is an example of how to factor a 33 matrix and keep tract of the
elementary row reduction matrices.
1
3 
2
4

6
6


 6  3  10
R2 = -2R1 + R2
1
3  2
1
3 
 1 0 0  2
  2 1 0  4
 0

6
6
4
0


 

 0 0 1  6  3  10  6  3  10
1
3  2 1 3 
1 0 0   2
0 1 0   0
  0 4 0 
4
0


 

3 0 1  6  3  10 0 0  1
R3 = 3R1 + R3
The matrix is now upper
triangular we combine
the two multiplications.
1
3  2 1 3 
1 0 0  1 0 0  2
0 1 0    2 1 0   4
  0 4 0 
6
6



 

3 0 1  0 0 1  6  3  10 0 0  1
Use matrix
arithmetic and
substitution.
1
 1 0 0
1 0 0 
  2 1 0   2 1 0




 0 0 1
0 0 1
1
1 0 0
 1 0 0
0 1 0    0 1 0 




3 0 1
 3 0 1
Calculating the inverse for elementary matrices
of this type can be done quickly (i.e. a shortcut)
by just taking the negative of the nonzero, nondiagonal entry.
1
1
1
3   1 0 0 1 0 0 2 1 3 
2
4
    2 1 0  0 1 0   0 4 0 
6
6

 
 
 

 6  3  10  0 0 1 3 0 1 0 0  1
1
3   1 0 0   1 0 0    2 1 3 
2


4





6
6

2
1
0
0
1
0
0
4
0






 



 6  3  10  0 0 1  3 0 1  0 0  1
1
3   1 0 0 2 1 3 
2
4
   2 1 0 0 4 0 
6
6

 


 6  3  10  3 0 1 0 0  1
Multiply by the
inverses from
previous slide.
Substitute and
group elementary
matrices
A short cut for multiplying
these matrices is putting the
entries in the corresponding
positions.
LU Factorization (Short Cut)
We do not need to do all the moving around of matrices on each side of the
equation. We just keep tract of the row operations and fill in the entries of the
corresponding matrix to get a sequence of lower triangular matrices.
3 2 0 
12 7 2 


 0 5  8
R2=-4R1+R2
0
3 2
0  1 2 


0 5  8
R3=5R2+R3
3 2 0 
0  1 2 


0 0 2
1 0 0 
L1  4 1 0
0 0 1
1 0 0 
L  4 1 0
0  5 1
Stop here the matrix is upper triangular
and this is the matrix U.
Can check that the
factorization is correct
with a matrix
multiplication
1
LU  4
0
3
 12
 0
0
1
5
2
7
5
0 3 2 0 
0 0  1 2
1 0 0 2
0
2 
 8
Matrices Without LU Factorizations
0 1
M 

1
0


det M  1
Not every matrix has an LU Factorization, just
like not all numbers or polynomials can be
factored. Look at the simple 22 matrix M given
to the right.
To show this is impossible to factor this way
consider what L and U would need to do.
L is lower triangular and U is upper triangular.
M = LU
This means that ae = 0 so that either a = 0 or
e=0
If a=0 then detL=0 so the detLU=0
If e=0 then detU=0 so the detLU=0
a 0 
L

c d 
a
LU  
c
0
M 
1
e
U 
0
f
h 
0  e
d  0
f  ae
af 

h   ce cf  dh
1
af 
ae

LU

 ce cf  dh
0


0 0 
0
L
or
U


0
c d 

When row reducing the matrix if you ever need to do the row operation of
interchanging (swapping) two rows the matrix can not be LU factored.
f
h 