Question

How can I vectorize this loop, which populates two square submatrices of a larger matrix (also keeps larger matrix symmetric) using numpy arrays:

for x in range(n):
    assert m[x].shape == (n,)
    M[i:i+n,j+x] = m[x]
    M[j+x,i:i+n] = m[x]

This is tempting but does not agree with loop above (see example disagreement below):

assert m.shape == (n,n)
M[i:i+n,j:j+n] = m
M[j:j+n,i:i+n] = m

Here’s a little example (crashes for n>1):

from numpy import arange,empty,NAN
from numpy.testing import assert_almost_equal

for n in (1,2,3,4):
    # make the submatrix
    m = (10 * arange(1, 1 + n * n)).reshape(n, n)

    N = n # example below, submatrix is the whole thing

    # M1 using loops, M2 "vectorized"
    M1 = empty((N, N))
    M2 = empty((N, N))
    M1.fill(NAN)
    M2.fill(NAN)

    i,j = 0,0 # not really used when (n == N)

    # this results in symmetric matrix
    for x in range(n):
        assert m[x].shape == (n,)
        M1[i:i+n,j+x] = m[x]
        M1[j+x,i:i+n] = m[x]

    # this does not work as expected
    M2[i:i+n,j:j+n] = m
    M2[j:j+n,i:i+n] = m

    assert_almost_equal(M1,M1.transpose(),err_msg="M not symmetric?")
    print "M1\n",M1,"\nM2",M2
    assert_almost_equal(M1,M2,err_msg="M1 (loop) disagrees with M2 (vectorized)")

We end up with:

M1 = [10 30
      30 40] # symmetric

M2 = [10 20
      30 40] # i.e. m

Answer 1

Your test is incorrect:
for i,j=0,0 your M2[]= assignments just overwrite the same matrix block.

The reason you get the symmetric matrix when using M1 is because you assign
the M1 values in a single loop.

if you would split the loop into two:

for x in range(n):
      M1[i:i+n,j+x] = m[x]
for x in range(n): 
      M1[j+x,i:i+n] = m[x]

The M1 will be obviously the same as M2.

So Summarizing, the following code works (equivalent to your M2 calculation) but ! it will only work if there is no overlap between blocks above and below the diagonal. if there is you have to decide what to do there

xs=np.arange(4).reshape(2,2)
ys=np.zeros((7,7))
ys[i:i+n,j:j+n]=xs
ys[j:j+n,i:i+n]=xs.T
print ys
>> array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  1.,  0.,  0.],
       [ 0.,  0.,  0.,  2.,  3.,  0.,  0.],
       [ 0.,  0.,  2.,  0.,  0.,  0.,  0.],
       [ 0.,  1.,  3.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]])