Let $v_1$ and $v_2$, with $v_1\neq v_2$ and $\langle v_1,v_2\rangle\neq 0$, be two non zero vectors in an $N$ dimensional Hilbert space. Let $\mathbf{S}=v_1\otimes v_1 +v_2\otimes v_2$ be an $N\times N$ Hermitian matrix ($\otimes$ denoting outer product).
$\mathbf{S}$ has at most 2 non-zero eigenvalues. Assuming there are 2 distinct eigenvalues and denoting the two associated eigenvectors as $e_1$ and $e_2$, can it be said with certainty that $\langle e_1, v_1\rangle\neq 0$ and $\langle e_1,v_2\rangle\neq 0$ (and similarly for $e_2$)?