ENH: @ operator differs from the convention for tensor dot

[WenyinWei]

Describe the issue:

Hi, the current @ operator has np.matmul function underlying, which means if $A$ or $B$ is multi-dimensional, C = A@B is equivalent to $C_{ijm} = \sum_{k} A_{ijk} B_{ikm}$. However, for multi-dimensional array, is would be more convenient and intuitive if it can somewhat imitate the behavior of tensor dots like using C = A@@B to calculate $C_{ij} = \sum_{m,n} A_{imn} B_{nmj}$. Is it possible to overload functions for such @@@... operators?

Reproduce the code example:

import numpy as np
testA, testB = np.random.rand(5,3,4), np.random.rand(5,4,3)
np.allclose( testA @ testB, np.einsum("ijk,ikm->ijm", testA, testB  )  )

Error message:

No response

Python and NumPy Versions:

1.26.4
3.11.9 (tags/v3.11.9:de54cf5, Apr 2 2024, 10:12:12) [MSC v.1938 64 bit (AMD64)]

Runtime Environment:

No response

Context for the issue:

No response

[ngoldbaum]

See numpy.vecdot. Of course that doesn't help someone wanting to use @ like you'd like, but changing it would also almost certainly break user code so there would need to be a deprecation period, a period where using @ in this way is a hard error, and then finally @ is an alias for vecdot. Is the change worth the disruption?

[seberg]

@ was proposed in the PEP to be useful for stacks of matrices, I don't think we should touch it and I would suspect that the stack of matrices use case is more common anyway.

Changing/adding operators in Python will not be easy, matmul was one of the few things that happened.

[ngoldbaum]

Also see np.einsum, which lets you write the contraction you want explicitly using Einstein notation. Closing since there's nothing really numpy can do to add more operators to Python and the operation the reporter was looking for is possible with a different spelling.