Smallest Eigenvector

Question

I am trying to show (or disprove) the following:

If v is the eigenvector corresponding to the smallest eigenvalue (in absolute value) of a square positive semidefinite matrix S, and

w = inverse(S) * 1

where 1 is the vector where each component is one, then

w is in the same direction as v.

Thank you

Answer 1

Let $$S=\begin{pmatrix}2&0\\0&1\end{pmatrix},$$ then $v= \begin{pmatrix}0\\1\end{pmatrix}$.

$$S^{-1}=\begin{pmatrix}1/2&0\\0&1\end{pmatrix},$$

$$w=S^{-1}\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}1/2\\1\end{pmatrix}.$$ Clearly, the hypothesis about the collinearity of $v$ and $w$ is false.