The following Gram-Schmidt orthogonalization procedure implementation is modified to use the city-block or manhattan-distance metric (instead of the typical Euclidean distance metric) so as to preserve the nature of any rational vectors input into the function.

gs.py

import numpy as np
np.seterr(divide='ignore', invalid='ignore')
from numpy import linalg
from functools import reduce
from matplotlib import pyplot as plt
import matplotlib.image as mpimg
import urllib.request

def norm1(v):
    """
    Returns the L1 (city-block, manhattan distance, taxicab) norm 
    """
    return np.sum(np.absolute(v))

def proj(v, G):
    """
    This function finds the projection of vector v onto the line, plane,
    hyperplane, etc. formed by the orthonormal vectors within G. Because we're 
    working with orthonormal bases, proj(v,g) = (v ⋅ g)g.

    Returns the projection of v onto g.
    """
    
    return (np.dot(v, G[0]) * G[0]) if len(G) == 1 else reduce(lambda a,b : (np.dot(v, a) * a) + (np.dot(v, b) * b), G)

def gs(W):
    """
    Takes in a finite set of vectors and generates an orthonormal basis.
    """
    G = []
    i = 1
    for w in W:
        if G == []:            
            g = np.divide(w, norm1(w))
            G.append(g)
        else:            
            g = w - proj(w, G)
            g = np.divide(g, norm1(g))
            G.append(g)
        print(i)
        i += 1
    return G

# 1. Load Jimmi Hendrix image
img = mpimg.imread("hendrix_final.png")
print(img.shape)

# 2. Extract red color channel (2000 x 2000 matrix)
red_img = (img[0:len(img), 0:len(img), 0])
plt.imshow(red_img)
plt.show()

# 3. Perform modified Gram-Schmidt on rows of image
reduced_red_img = gs(red_img)

# 4. Show reduced_red_img
plt.imshow(reduced_red_img)