How to understand the pivot matrix of scipy.linalg.lu_factor?

The permutation vector needs to be interpreted in sequence. If piv=[1,2,2] then the following needs to be done in sequence (with zero-based indexing):

1. Row 0 changes with Row 1
2. The new Row 1 changes with Row 2 and
3. The new Row 2 stays the same.

In code this would do the trick:

P = np.eye(3)
for i, p in enumerate(piv):
    Q = np.eye(3,3)
    q = Q[i,:].copy()
    Q[i,:] = Q[p,:]
    Q[p,:] = q
    P = np.dot(P, Q)

For piv=[1,2,2] P is

[[ 0.  0.  1.]
 [ 1.  0.  0.]
 [ 0.  1.  0.]]

This is probably not a very fast way of computing P but it does the trick and answers the question.

Or with slight refactoring -

def constructPermutationMatrixFromPermutationVector(piv):
        n = len(piv)
        P = np.eye(n)
        for kk, ll in enumerate(piv):
            P[:, kk], P[:, ll] = P[:, ll].copy(), P[:, kk].copy()
        return P