My class has recently learnt the BFGS method for unconstrained optimisation. In this procedure, we have a rank-1 update to a positive definite matrix at each step.
This is specified as:
$H_{k+1} = H_k + \frac{\eta\eta^T}{\delta^T\eta}-\frac{H_k\delta\delta^TH_k^T}{\delta^TH_k\delta}$
$\forall \eta, \delta \in \mathbb{R}^n$.
Show that for any symmetric positive definite matrix $H_k,$ we have that $H_{k+1}$ is positive definite so long as $\delta^T\eta > 0$. Don't assume anything about $H_k$ other than the fact that it is symmetric p.d.