Scipy LU factorization permutation matrix

Question

As I understand it, LU factorization means that the matrix A can be written as A = LU for the lower triangular matrix L and the upper triangular matrix U.

However, the functions in scipy, relating to the LU-decompositions ( lu, lu_factor, lu_solve), it appears includes a third matrix P, such that A = PLU and P is a permutation matrix (and L, U remains).

What is the meaning of this permutation matrix? If “true” LU factorization is always possible, why has P ever been anything other than an identity matrix?

Answer 1

. , ? , P.

Moreso, . .

, , , . 10, . , , .

" LU " .

. P - , P ^ T = P ^ (- 1). , Ax = b , LUx = P ^ T b ( , P, , P ^ T, P - , , "PA = LU" "A = PLU" - P ).

Answer 2

LU. LU.

Answer 3

@DomJack: (aka reordering) L U . , , .