LU factorization for rectangular matrix seems incomplete

The following example shows 3x4 maxtrix lu factorization in MATLAB seems incomplete.

MATLAB's answer for u is [1 3 3 2; 0 0 3 3; 0 0 6 6].

The answer in the text book (Gilbert Strang) is [1 3 3 2; 0 0 3 3; 0 0 0 0].

Is thera a way to get the same answer as the one from the text book through MATLAB?

----------------------------------------------------------------------------------------------------------
 a =
 
[  1,  3, 3, 2]
[  2,  6, 9, 7]
[ -1, -3, 3, 4]
 
>> [l,u,p]=lu(a)
 
l =
 
[  1, 0, 0]
[  2, 1, 0]
[ -1, 0, 1]
 
 
u =
 
[ 1, 3, 3, 2]
[ 0, 0, 3, 3]
[ 0, 0, 6, 6]
 

p =

     1     0     0
     0     1     0
     0     0     1

[Bruno Luong]

Are you runing MATLAB or something else? Your formatting seems not to be MATLAB and your result does not match mine (R2018a).

a

a =

     1     3     3     2
     2     6     9     7
    -1    -3     3     4

>>  [l,u,p]=lu(a)

l =

   1.000000000000000                   0                   0
   0.500000000000000   1.000000000000000                   0
  -0.500000000000000                   0   1.000000000000000


u =

   2.000000000000000   6.000000000000000   9.000000000000000   7.000000000000000
                   0                   0  -1.500000000000000  -1.500000000000000
                   0                   0   7.500000000000000   7.500000000000000


p =

     0     1     0
     1     0     0
     0     0     1

[Soo Chang Choe]

run 'a = sym(a);' before lu() function. You will get the same format.

However, your numerical results also have the same problem. U's last row should be [0 0 0 0], but it is not.

[Bruno Luong]

Why there is "problem" or "incomplete"? It's upper triangular as advertised.

For your information LU factorization is not unique.

[Soo Chang Choe]

Mathematically it is not a problem or incomplete. But is there a built-in matlab funciton that gives me the same answer as can be found in the text book? I don't understand why it stopped pivoting at the second row.

[Bruno Luong]

"I don't understand why it stopped pivoting at the second row."

It does NOT stop, at the second step it pivots at the diagonal elements which is 0 for both row 2 and 3 (draw) so it picks row 2. Elements above/on the right the diagonal of u are not supposed to play any role.

In your book it selects row 3, but it is just a choise, without any additional usefulness for what it design for: solving linear system.

Both are equally valid.

[Soo Chang Choe]

Hello Bruno,

I wanted to use MATLAB to check my answer when studying the text book.

It seems that I have to build my own funciton to get the answer that the book expects for now.

As you said, I am convinced that MATLAB is generating valid answers for LU decomposition.

Many thanks for your reply and answer.