Compute a sum in numpy array

Question

I have an m by n matrix A, implemented as a numpy array.

import numpy as np
m = 10
n = 7
A = np.random.rand(m, n)

I want to compute the m by m matrix B whose entries are

B[i, j] = sum_{k=1,...,n} sum_{l=1,...,n} A[i, k] * A[j, l]

What is the easiest way to do this without making explicit for loops?

@SimonParker the code in your edit does not compute sum_{k} sum_{l} A[i, k] * A[j, l], but sum_{k} A[i, k] * A[j, k], which @Divakar's A.dot(A.T) gives you. — Alicia Garcia-Raboso
– Alicia Garcia-Raboso, Commented May 6, 2018 at 8:50
@AlbertoGarcia-Raboso That's a mistake. The first version was the good one. I will erase it. — Simon Parker
– Simon Parker, Commented May 6, 2018 at 8:51

Alicia Garcia-Raboso · Accepted Answer · 2018-05-06 08:45:14Z

2

Notice that the sum over k in your expression only affects the first factor, while the sum over l only involves the second:

sum_{k=1,...,n} sum_{l=1,...,n} A[i, k] * A[j, l] =
    (sum_{k=1,...,n}  A[i, k]) * (sum_{l=1,...,n} A[j, l])

The expressions in parentheses are, except for the names of the indices, the same, so define

sA = np.sum(A, axis=1)

Then your B is the so-called outer product of sA with itself:

B = np.outer(sA, sA)

answered May 6, 2018 at 8:45

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