Let $A\in\mathbb{R}^{d\times d}$ be a symmetric, rank deficient matrix ($\text{rank}(A)< d$). Let $D\in\mathbb{R}^{d\times d}$ be another matrix that is diagonal, with possibly negative entries on the diagonal.
I want to lower bound $\|ADA\|_{op}$, where $\|\cdot\|_{op}$ denotes the operator norm/largest singular value.
In particular, I want something like $\|A\|_{op}^{2}$ to figure in the lower bound.
Let $d_{min}:=min_{i}D_{ii}$. I tried to proceed by using the fact that $ADA\succeq A(d_{min}I)A$, which gives $\|ADA\|_{op}\ge d_{min}\|A\|_{op}^{2}$, but since $d_{min}$ can be negative, this is not useful.
Any hints appreciated.