Does $u v^T + v u^T$ have exactly one positive and one negative eigenvalue when $u \not \propto v$?
$u$ and $v$ are column vectors in $\mathbb{R}^n$.
Does $u v^T + v u^T$ have exactly one positive and one negative eigenvalue when $u \not \propto v$?
$u$ and $v$ are column vectors in $\mathbb{R}^n$.
I presume that $u$, $v$ are nonzero. By absorbing the length of $u$ into $v$, we may assume WLOG that $\|u\|=1$. Let $v=\alpha u + \beta w$ for some unit vector $w\perp u$. Note that $\beta\not=0$ because $u \not \propto v$. Also, $uv^\top+vu^\top = 2\alpha uu^\top+\beta uw^\top+\beta wu^\top$. Hence $ uv^\top+vu^\top \sim \begin{pmatrix} 2\alpha&\beta\\ \beta&0 \end{pmatrix}\oplus0_{n-2} $ and the result follows immediately.