x
Out[25]:
<11250x11250 sparse matrix of type '<class 'numpy.float64'>'
    with 44547 stored elements in Compressed Sparse Column format>

g = np.zeros((x.shape[0]))
g[idx] = 0.1

from scipy.sparse.linalg import spsolve
F = spsolve(x, g)

scipy.sparse.linalg.dsolve.linsolve.MatrixRankWarning: Matrix is exactly singular

spsolve(x* 1.00000001, g)
Out[53]: array([-0., -0., -0., ..., -0., -0., -0.])
spsolve(x/ 1.00000001, g)
Out[54]: array([-0., -0., -0., ..., -0., -0., -0.])

A.T g = z