Let $A \in \mathbb{R}_{+}^{n \times n}$ be diagonal with $i$th diagonal element $a_{ii}$ (i.e. all $a_{ii} > 0$), and let $X \in \mathbb{R}^{n \times p}$ be a matrix with rank $p$ where $n \ge p$ (in particular, consider a design matrix for regression).
Consider the matrix $X^{T} A X$. I'm interested in finding upper bounds on the matrix's largest eigenvalue, and lower bounds on the matrix's smallest eigenvalue. Alternatively, if $v \in \mathbb{R}^{p}$, I would be interested in upper and lower bounds for $v^{T} X^{T} A X v$. Ideally these bounds would relate to the matrix $X^{T} X$, or $\lambda_{max}(X^{T} X)$ and $\lambda_{min}(X^{T}X)$. It's clear that
$$ v^{T} X^{T} A X v \le \underset{i = 1, \ldots, p}{max} [a_{ii}] \cdot v^{T} X^{T} X v \le \left( \sum_{i=1}^{n} a_{ii} \right) \cdot v^{T} X^{T} X v \le \left( \sum_{i=1}^{n} a_{ii} \right) \cdot \lambda_{max}(X^{T} X) \cdot v^{T} v. $$
It's also relatively clear that
$$ v^{T} X^{T} A X v \ge \underset{i = 1, \ldots, p}{min} [a_{ii}] \cdot v^{T} X^{T} X v \ge \underset{i = 1, \ldots, p}{min} [a_{ii}] \cdot \lambda_{min}(X^{T} X) \cdot v^T v.$$
It would be ideal for me to have some type of lower bound on $v^{T} X^{T} A X v$ that relates to the trace of $A$. Does anyone have any ideas for how to achieve such a bound? Thanks so much for your help!