How to understand the matmul function when matrix a is two-dimensional and matrix b is three-dimensional?

a=np.arange(8).reshape(2,2,2)
b=np.arange(4).reshape(2,2)
print(np.matmul(a,b))

the Result is:

[[[ 2  3]
  [ 6 11]]

 [[10 19]
  [14 27]]]

I don't understand this result, can someone please explain it?

Short answer: it "broadcasts" the second 2d matrix to a 3d matrix, and then performs a "mapping" so, it maps the elementwise submatrices to new submatrices in the result.

As the documentation on np.matmul [numpy-doc] says:

numpy.matmul(a, b, out=None)

Matrix product of two arrays.

The behavior depends on the arguments in the following way.

1. If both arguments are 2-D they are multiplied like conventional matrices.
2. If either argument is N-D, N > 2, it is treated as a stack of matrices residing in the last two indexes and broadcast accordingly.
3. If the first argument is 1-D, it is promoted to a matrix by prepending a 1 to its dimensions. After matrix multiplication the prepended 1 is removed.
4. If the second argument is 1-D, it is promoted to a matrix by appending a 1 to its dimensions. After matrix multiplication the appended 1 is removed.

So here the second item is applicable. So first the second matrix is "broadcasted" to the 3d variant as well, so that means that we multiple:

array([[[0, 1],
        [2, 3]],

       [[4, 5],
        [6, 7]]])

with:

array([[[0, 1],
        [2, 3]],

       [[0, 1],
        [2, 3]]])

and we see these as stacked matrices. So first we multiply:

array([[0, 1],      array([[0, 1],
       [2, 3]])  x        [2, 3]])

which gives us:

array([[ 2,  3],
       [ 6, 11]])

and then the elementwise second submatrices:

array([[4, 5],      array([[0, 1],
       [6, 7]])  x        [2, 3]])

an this gives us:

array([[10, 19],
       [14, 27]])

we thus stack these together into the result, and obtain:

>>> np.matmul(a, b)
array([[[ 2,  3],
        [ 6, 11]],

       [[10, 19],
        [14, 27]]])

Although the behavior is thus perfectly defined, it might be better to use this feature carefully, since there are other "sensical" definitions of what a "matrix product" on 3d matrices with 2d matrices might look like, and these are thus not used here.