What is the interpretation of the eigenvectors of the jacobian matrix? I'm trying to think about the jacobian matrix as a abstract linear map.
What is the interpretation of the eigenvalues and eigenvectors of the jacobian?

What Frank Science has said in the question comment above is right. I'm simply expanding on his comment here:
Since the Jacobian has eigenvectors, it is square i.e. the input and output space have same dimensions. If there is some kind of natural interpretation to the input and output basis and they can be mapped to each other, the eigenvectors of the Jacobian represent those directions in the input space, where if you move locally (small amount), you move in the same direction in the output space. This interpretation is no different from that of the eigenvectors of any matrix. The only addition is that of local (small) movement, because the Jacobian approximates the original function locally.
