How to generate a Rank 5 matrix with entries Uniform?

Question

I want to generate a rank 5 100x600 matrix in numpy with all the entries sampled from np.random.uniform(0, 20), so that all the entries will be uniformly distributed between [0, 20). What will be the best way to do so in python?

I see there is an SVD-inspired way to do so here (https://math.stackexchange.com/questions/3567510/how-to-generate-a-rank-r-matrix-with-entries-uniform), but I am not sure how to code it up. I am looking for a working example of this SVD-inspired way to get uniformly distributed entries.

I have actually managed to code up a rank 5 100x100 matrix by vertically stacking five 20x100 rank 1 matrices, then shuffling the vertical indices. However, the resulting 100x100 matrix does not have uniformly distributed entries [0, 20).

Here is my code (my best attempt):

import numpy as np
def randomMatrix(m, n, p, q):
    # creates an m x n matrix with lower bound p and upper bound q, randomly.
    count = np.random.uniform(p, q, size=(m, n))
    return count

Qs = []
my_rank = 5
for i in range(my_rank):
  L = randomMatrix(20, 1, 0, np.sqrt(20))
  # L is tall
  R = randomMatrix(1, 100, 0, np.sqrt(20)) 
  # R is long
  Q = np.outer(L, R)
  Qs.append(Q)

Q = np.vstack(Qs)
#shuffle (preserves rank 5 [confirmed])
np.random.shuffle(Q)

Answer 1

Not a perfect solution, I must admit. But it's simple and comes pretty close.
I create 5 vectors that are gonna span the space of the matrix and create random linear combinations to fill the rest of the matrix. My initial thought was that a trivial solution will be to copy those vectors 20 times.
To improve that, I created linear combinations of them with weights drawn from a uniform distribution, but then the distribution of the entries in the matrix becomes normal because the weighted mean basically causes the central limit theorm to take effect.
A middle point between the trivial approach and the second approach that doesn't work is to use sets of weights that favor one of the vectors over the others. And you can generate these sorts of weight vectors by passing any vector through the softmax function with an appropriately high temperature parameter.
The distribution is almost uniform, but the vectors are still very close to the base vectors. You can play with the temperature parameter to find a sweet spot that suits your purpose.

from scipy.stats import ortho_group
from scipy.special import softmax
import numpy as np
from matplotlib import pyplot as plt
N    = 100
R    = 5
low  = 0
high = 20
sm_temperature = 100

p       = np.random.uniform(low, high, (1, R, N))
weights = np.random.uniform(0, 1, (N-R, R, 1))
weights = softmax(weights*sm_temperature, axis = 1)
p_lc    = (weights*p).sum(1)

rand_mat = np.concatenate([p[0], p_lc])

plt.hist(rand_mat.flatten())