# LU decomposition: lu

### 5.49.5 LU decomposition: lu

The LU decomposition of a square matrix A is P A= L U, where P is a permutation matrix, L is lower triangular with 1s on the diagonal, and U is upper triangular. The lu command finds the LU decomposition of a matrix.

- lu takes one argument:
  A, a square matrix.
- lu(A) returns a list [p,L,U] where p is a permutation that determines P, and P, L and U are the LU decomposition of A.

The permutation matrix P is defined from p by:

Pi,p(i)=1, Pi,j=0 if j ≠ p(i)

In other words, it is the identity matrix where the rows are permuted according to the permutation p. You can get the permutation matrix from p by P:=permu2mat(p) (see Section 5.9.6).

#### Example.

Input:

A := [[3.,5.],[4.,5.]]:;
(p,L,U):=lu(A)

Output:

⎡
⎣

1,0

⎤
⎦

,

⎡
⎢
⎣

1 | 0
0.75 | 1

⎤
⎥
⎦

,

⎡
⎢
⎣

4.0 | 5.0
0 | 1.25

⎤
⎥
⎦

Here n=2, hence:

P[0,p(0)]=P2[0,1]=1, P[1,p(1)]=P2[1,0]=1, P=[[0,1],[1,0]]

Verification:
Input:

permu2mat(p)*A; L*U

Output:

⎡
⎢
⎣

4.0 | 5.0
3.0 | 5.0

⎤
⎥
⎦

,

⎡
⎢
⎣

4.0 | 5.0
3.0 | 5.0

⎤
⎥
⎦

Note that the permutation is different for exact input (the choice of pivot is the simplest instead of the largest in absolute value).
Input:

lu([[1,2],[3,4]])

Output:

⎡
⎣

0,1

⎤
⎦

,

⎡
⎢
⎣

1 | 0
3 | 1

⎤
⎥
⎦

,

⎡
⎢
⎣

1 | 2
0 | −2

⎤
⎥
⎦

Input:

lu([[1.0,2],[3,4]])

Output:

⎡
⎣

1,0

⎤
⎦

,

⎡
⎢
⎣

1 | 0
0.333333333333 | 1

⎤
⎥
⎦

,

⎡
⎢
⎣

3.0 | 4.0
0 | 0.666666666667

⎤
⎥
⎦