LU Decoposition

It is a matrix factorization method where it factors a square matrix into Lower and Upper triangular matrix L and U such that A=L U The primary importance of LU decomposition is to solve the linear systems AX=Y, perticularly, when solving same A but different Y. The matrix L result from clearing all the values above the main diagonal via Gaussian Elimination. The matrix U reflects what row operations have occurred in the course of building L.Simplifying U and applying L to Y requires less work than solving A from scratch. Also, LU decomposition help us to determine the determinant of Matrix. The product of Diagonal of U is a determinant. Look at the grayscale image of 64x64 pixels. we'll use scipy and numpy to decompose it in L and U and the permutation matrix P.

from PIL import Image
import numpy as np
from scipy.linalg import lu
from matplotlib import pyplot as plt

img=Image.open('gray64x64.png')
A=np.array(img)
P,L,U=lu(A)

#plt.imshow(L,interpolation='nearest')
#plt.imshow(U,interpolation='nearest')
#plt.imshow(P@L@U,interpolation='nearest')
plt.show()

This is How it looks when line 16 is uncommendted.

This is How it looks when line 17 is uncommendted.

You can see above that without appliying permutation, we've bottom lines shifted to top. This is How it looks when line 18 is uncommendted.