Eigenvalues of an outer product

Question

Let $w, v \in \mathbb{R}^d$. What is known about the eigenvalues of the outer product matrix $vw^{\top} + wv^{\top}$?

Answer 1

If $w,v$ are colinear, the outer product is $2v w^T $, which has $0$ as an eigenvalue $d-1$ times and $v^T w = v \cdot w$ as its nonzero eigenvalue.

Otherwise, $w,v$ span a two-dimensional subspace. The orthogonal complement of this space is the kernel, so $0$ is an eigenvalue $d-2$ times. On the subspace spanned by $w,v$, with respect to that basis the outer product has the matrix $$ \begin{pmatrix} w \cdot v & w \cdot w \\ v \cdot v & w \cdot v \end{pmatrix} , $$ from which the eigenvalues follow in the usual way, and are $$ w \cdot v \pm \sqrt{ ( w \cdot w)( v \cdot v) } . $$ By the triangle inequality, one of these is always positive and one negative.