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I have a matrix M with values 0 through N within it. I'd like to unroll this matrix to create a new matrix A where each submatrix A[i, :, :] represents whether or not M == i.

The solution below uses a loop.

# Example Setup
import numpy as np

np.random.seed(0)
N = 5
M = np.random.randint(0, N, size=(5,5))

# Solution with Loop
A = np.zeros((N, M.shape[0], M.shape[1]))
for i in range(N):
 A[i, :, :] = M == i

This yields:

M
array([[4, 0, 3, 3, 3],
 [1, 3, 2, 4, 0],
 [0, 4, 2, 1, 0],
 [1, 1, 0, 1, 4],
 [3, 0, 3, 0, 2]])

M.shape
# (5, 5)


A 
array([[[0, 1, 0, 0, 0],
 [0, 0, 0, 0, 1],
 [1, 0, 0, 0, 1],
 [0, 0, 1, 0, 0],
 [0, 1, 0, 1, 0]],
 ...
 [[1, 0, 0, 0, 0],
 [0, 0, 0, 1, 0],
 [0, 1, 0, 0, 0],
 [0, 0, 0, 0, 1],
 [0, 0, 0, 0, 0]]])

A.shape
# (5, 5, 5)

Is there a faster way, or a way to do it in a single numpy operation?

Broadcasted comparison is your friend:

B = (M[None, :] == np.arange(N)[:, None, None]).view(np.int8)

 np.array_equal(A, B)
# True

The idea is to expand the dimensions in such a way that the comparison can be broadcasted in the manner desired.

As pointed out by @Alex Riley in the comments, you can use np.equal.outer to avoid having to do the indexing stuff yourself,

B = np.equal.outer(np.arange(N), M).view(np.int8)

np.array_equal(A, B)
# True

You can make use of some broadcasting here:

P = np.arange(N)
Y = np.broadcast_to(P[:, None], M.shape)
T = np.equal(M, Y[:, None]).astype(int)

Alternative using indices:

X, Y = np.indices(M.shape)
Z = np.equal(M, X[:, None]).astype(int)

You can index into the identity matrix like so

 A = np.identity(N, int)[:, M]

or so

 A = np.identity(N, int)[M.T].T

Or use the new (v1.15.0) put_along_axis

A = np.zeros((N,5,5), int)
np.put_along_axis(A, M[None], 1, 0)

Note if N is much larger than 5 then creating an NxN identity matrix may be considered wasteful. We can mitigate this using stride tricks:

def read_only_identity(N, dtype=float):
 z = np.zeros(2*N-1, dtype)
 s, = z.strides
 z[N-1] = 1
 return np.lib.stride_tricks.as_strided(z[N-1:], (N, N), (-s, s))