Currently the linalg extension only contains symmetric/hermitian eigenvalue routines, eigh and eigvalsh. The general/non-symmetric versions are missing.
IIUC one reason for the omission is that the low-level routines were not universally available on CUDA or were of allegedly questionable quality [1].
Now that the needed LAPACK-like routine was added to cuSolver 12.6 update 2 [2], the low-level blocker is no more.

Availability:

• numpy

np.linalg.eig, np.linalg.eigvals are available: https://numpy.org/doc/2.1/reference/generated/numpy.linalg.eig.html

• torch

torch.linalg.eig, torch.linalg.eigvals are available: https://pytorch.org/docs/stable/generated/torch.linalg.eig.html

• jax

jax.numpy.linalg.eig, jax.numpy.linalg.eigvals are available: https://docs.jax.dev/en/latest/_autosummary/jax.numpy.linalg.eig.html

Under the hood, it uses the same MAGMA kernels as pytorch, so [1] applies. However, jax-ml/jax#27265 tracks a potential update to use the new routines from [2].

• cupy

Not available in CuPy 13.4 yet; There is a pull request to add it: cupy/cupy#8980, hopefully to CuPy 14.x.

Proposed API

The API should closely follow eigh and eigvalsh:

• eigvals(x, /) should return an array of eigenvalues in an unspecified order;

• eig(x, /) should return a namedtuple (eigenvalues, right eigenvectors). The order of eigenvalues and eigenvectors is unspecified, and the only requirement is that eigenvalues[i] corresponds to eigenvectors[:, i].

Whether the eigenvectors have a unit norm is unspecified and thus implementation-defined.

An option to return left eigenvectors may be added in the future if the array libraries add this option (currently, neither numpy nor torch provide this option).

[1] cupy/cupy#6359 (comment)
[2] https://docs.nvidia.com/cuda/cuda-toolkit-release-notes/index.html