Why does np.linalg.eig produce an opposite-signed eigenvector?
Any scalar multiple of an eigenvector is also an eigenvector.  LAPACK (which np.linalg.eig uses under the hood) chooses to return unit-length eigenvectors (good for SVD!), but this still leaves two choices, and there doesn't seem to be a convention for which one to return; it's up to the underlying algorithm (which in turn may depend on the input data).

https://stackoverflow.com/questions/17998228/sign-of-eigenvectors-change-depending-on-specification-of-the-symmetric-argument