How can I compute a generalized Kronecker product using XOR in NumPy?

Below is one acceptable solution. We “expand” A so that each element of A is repeated in a block of the same shape as B, then we apply the XOR (using np.logical_xor for booleans) with a tiled version of B. For example:

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import numpy as np

A = np.asarray([[True,  False],
                [False, False]])

B = np.asarray([[False, True],
                [True,  True]])

# Repeat each element of A to form blocks of size B.shape:
A_expanded = A.repeat(B.shape[0], axis=0).repeat(B.shape[1], axis=1)
# Tile B so that it fits in the full kron-like matrix:
B_tiled = np.tile(B, A.shape)

# Perform elementwise XOR on the two:
C = np.logical_xor(A_expanded, B_tiled)
print(C)

This produces:

[[ True False False  True]
 [False False  True  True]
 [False  True False  True]
 [ True  True  True  True]]

A More General Approach

If you want a “Kronecker product” type operation where the inner multiplication is replaced by another binary operator (here XOR), you can write a helper function. For example:

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def kron_op(A, B, op):
    """Compute a generalized Kronecker product with binary operator op."""
    m, n = A.shape
    p, q = B.shape
    # Use np.block to build the result from blocks:
    return np.block([[ op(A[i, j], B) for j in range(n)]
                     for i in range(m)])

C2 = kron_op(A, B, np.logical_xor)
print(C2)

This gives the same result.

Both methods are simple and take advantage of NumPy’s broadcasting and array manipulation functions.