函数scipy.linalg.lu中的上三角矩阵是否总是呈行梯形形式？

Question

I have amxn matrix A, with n > m, and I am trying to identify independent rows by means of the row echelon form of it. Function scipy.linalg.lu returns a PLU factorization of my matrix, but U factor does not seem to be in echelon form, ie, pivots are not in a staircase pattern. As far as I know, U factor should always be in a staircase pattern.

Consider the following example:

from numpy import array
from scipy.linalg import lu

A = array([[1, 1, 1, 1, 0, 1, 1, 1, 1, 0],
           [1, 1, 0, 0, 1, 0, 1, 0, 1, 1],
           [1, 1, 0, 0, 0, 1, 1, 0, 0, 0],
           [0, 1, 0, 1, 1, 0, 1, 0, 0, 1],
           [1, 1, 0, 0, 1, 1, 1, 1, 1, 1]])

P, L, U = lu(A)

The U factor is not in row echelon form. For each row k, pivots should always be to the right of the pivot in row k-1. See that pivot in fifth row is not to the right of pivot in fourth row:

array([[ 1.,  1.,  1.,  1.,  0.,  1.,  1.,  1.,  1.,  0.],
       [ 0.,  1.,  0.,  1.,  1.,  0.,  1.,  0.,  0.,  1.],
       [ 0.,  0., -1., -1.,  0.,  0.,  0., -1., -1.,  0.],
       [ 0.,  0.,  0.,  0.,  1., -1.,  0.,  0.,  1.,  1.],
       [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  1.,  1.,  1.]])

Answer 1

I've never seen the LUP decomposition before but I suspect it just doesn't do exactly what you think it does. I did the same decomposition in Matlab as you did above in Python and I got the exact same results you did. So I don't think it as problem with the Python LU function.

Update: I have also implemented this in R (code below) and again the U matrix gives the same results as Matlab and Python. Unlike the others it gives the following warning:

Warning message:
In .local(x, ...) :
  Exact singularity detected during LU decomposition: U[i,i]=0, i=4.

Matlab:

Code:

A = ([[1, 1, 1, 1, 0, 1, 1, 1, 1, 0],
           [1, 1, 0, 0, 1, 0, 1, 0, 1, 1],
           [1, 1, 0, 0, 0, 1, 1, 0, 0, 0],
           [0, 1, 0, 1, 1, 0, 1, 0, 0, 1],
           [1, 1, 0, 0, 1, 1, 1, 1, 1, 1]]);
[L,U,P] = lu(A);
U

Matlab results:
U =

     1     1     1     1     0     1     1     1     1     0
     0     1     0     1     1     0     1     0     0     1
     0     0    -1    -1     0     0     0    -1    -1     0
     0     0     0     0     1    -1     0     0     1     1
     0     0     0     0     1     0     0     1     1     1

Python Results (from your code):

U = 

array([[ 1.,  1.,  1.,  1.,  0.,  1.,  1.,  1.,  1.,  0.],
       [ 0.,  1.,  0.,  1.,  1.,  0.,  1.,  0.,  0.,  1.],
       [ 0.,  0., -1., -1.,  0.,  0.,  0., -1., -1.,  0.],
       [ 0.,  0.,  0.,  0.,  1., -1.,  0.,  0.,  1.,  1.],
       [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  1.,  1.,  1.]])

R:

Code:

library(Matrix)
A <- Matrix( c( 1, 1, 1, 1, 0, 1, 1, 1, 1, 0 , 1, 1, 0, 0, 1, 0, 1, 0, 1, 1 , 1, 1, 0, 0, 0, 1, 1, 0, 0, 0 , 0, 1, 0, 1, 1, 0, 1, 0, 0, 1 , 1, 1, 0, 0, 1, 1, 1, 1, 1, 1 ),nrow=5,ncol=10,byrow=TRUE )
expand(mylu <-lu(A))

Results:

class "dtrMatrix" (unitriangular)
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    .    .    .    .
[2,]    0    1    .    .    .
[3,]    1    0    1    .    .
[4,]    1    0    1    1    .
[5,]    1    0    1    0    1

$U
5 x 10 Matrix of class "dgeMatrix"
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    1    1    1    1    0    1    1    1    1     0
[2,]    0    1    0    1    1    0    1    0    0     1
[3,]    0    0   -1   -1    0    0    0   -1   -1     0
[4,]    0    0    0    0    1   -1    0    0    1     1
[5,]    0    0    0    0    1    0    0    1    1     1

$P
5 x 5 sparse Matrix of class "pMatrix"

[1,] | . . . .
[2,] . . . | .
[3,] . . | . .
[4,] . | . . .
[5,] . . . . |

Answer 2

There is no guarantee that the LU decomposition will result in a U matrix in row echelon form. It may be that the reason why it fails in this case is because the matrix is singular; I strongly suspect that is the reason, because one of the characteristics of a singular matrix is the absence of a complete set of pivots.

See the section on General Matrices at https://en.wikipedia.org/wiki/LU_decomposition and the reference provided therein for specifics.