How to define a matrix by vectorization (without for loop) in numpy?

Question

I want to define an NxN matrix A, whose element A(i,j) is sin(i^2 + j).

In MATLAB, I can quickly define this matrix by employing vectorization.

N = 2000;
ii = 1:N;
A = sin(ii.^2 + ii');

How to achieve this in Python? Right now, I am using for loops, which are slow.

import numpy
N = 2000;
A = numpy.empty((N, N));
for i in range(1,N+1): 
    for j in range(1,N+1): 
        A[i-1][j-1] = numpy.sin(j + numpy.power(i, 2.))

Answer 1

Many things you can do in MATLAB have exact translations in numpy. Here's a handy cheat-sheet

N = 2000 # remains unchanged
ii = np.arange(1, N+1)  # In the cheat-sheet section Linear Algebra Equivalents 

Now, there's a slight difference. In MATLAB, everything is treated as a matrix, so a 1-dimensional array is just a 1xN matrix, so transposing it is done just like any other matrix. In numpy, 1-dimensional arrays can't be transposed, so let's add a dummy dimension to ii:

ii = ii[None, :]

Now, ii is a 1xN matrix, so we can get A as:

A = np.sin(ii**2 + ii.T);

Numpy takes care of broadcasting the shapes (1, N) and (N, 1) and gives you a result that is (N, N)

^{Note that a' in MATLAB is the conjugate transpose, which would be a.conj().T in numpy, but since these are all real numbers it makes no difference}