Why does np.linalg.eig produce an opposite-signed eigenvector?

Question

I am learning SVD by following this MIT course. In this video, the lecturer is finding the SVD for

$$ \begin{pmatrix} 5 & 5 \\ -1 & 7 \end{pmatrix}, $$

which involves finding the eigenvalues for

$$ C^T C = \begin{pmatrix} 26 & 18 \\ 18 & 74 \end{pmatrix}. $$

In the example (at the time in the link above), the lecturer finds eigenvalues

$$\begin{pmatrix}-3/\sqrt{10} \\ 1/\sqrt{10} \end{pmatrix}, \begin{pmatrix} 1/\sqrt{10} \\ 3/\sqrt{10} \end{pmatrix}.$$

But np.linalg.eig produces the opposite vector to the second one:

w, v = np.linalg.eig(C.T*C)
v
matrix([[-0.9486833 , -0.31622777],
        [ 0.31622777, -0.9486833 ]])

Why?

Answer 1

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