How to make a simple Vandermonde matrix with numpy?

Question

My question is how to make a vandermonde matrix. This is the definition: In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an m × n matrix

I would like to make a 4*4 version of this.

So farI have defined values but only for one row as follows

a=2
n=4

for a in range(n):
    
    for i in range(n):
        v.append(a**i)
v = np.array(v)
print(v)

I dont know how to scale this. Please help!

Answer 1

Given a starting column a of length m you can create a Vandermonde matrix v with n columns a**0 to a**(n-1)like so:

import numpy as np

m = 4
n = 4

a = range(1, m+1)
v = np.array([a]*n).T**range(n)
print(v)
#[[ 1  1  1  1]
# [ 1  2  4  8]
# [ 1  3  9 27]
# [ 1  4 16 64]]

Answer 2

As proposed by michael szczesny you could use numpy.vander. But this will not be according to the definition on Wikipedia.

x = np.array([1, 2, 3, 5])
N = 4
np.vander(x, N)

#array([[  1,   1,   1,   1],
#       [  8,   4,   2,   1],
#       [ 27,   9,   3,   1],
#       [125,  25,   5,   1]])

So, you'd have to use numpy.fliplr aswell:

x = np.array([1, 2, 3, 5])
N = 4
np.fliplr(np.vander(x, N))

#array([[  1,   1,   1,   1],
#       [  1,   2,   4,   8],
#       [  1,   3,   9,  27],
#       [  1,   5,  25, 125]])

This could also be achieved without numpy using nested list comprehensions:

x = [1, 2, 3, 5]
N = 4
[[xi**i for i in range(N)] for xi in x]

#   [[1, 1, 1, 1],
#    [1, 2, 4, 8],
#    [1, 3, 9, 27],
#    [1, 5, 25, 125]]