If $x$ and $y$ are two linearly independent column $n$-vectors find all the eigenvalues of $xx^{T}-yy^{T}$

Question

If $x$ and $y$ are two linearly independent column $n$-vectors where $n\geq2$ .find all the eigenvalues of $xx^{T}-yy^{T}$

Answer 1

Hint: The matrix $xx^T-yy^T$ has rank $2$. So $n-2$ of the eigenvalues are $0$.

The other two eigenvectors have to lie in the columnspace of $xx^T-yy^T$, which is $\text{span}\{x,y\}$. So suppose $z = \alpha x + \beta y$ is an eigenvector of $xx^T-yy^T$ for some constants $\alpha$ and $\beta$. Can you find $\alpha$ and $\beta$ such that $(xx^T-yy^T)z = \lambda z$?