Consider the following linear matrix inequality (LMI)
$$ A - \delta B^{\text{T}} B > 0 \tag{1} \label{1} $$
where $ A \in \mathbb{R}^{n \times n} $ is a known symmetric positive definite matrix, $ B \in \mathbb{R}^{m \times n} $ is also known, and $ \delta \in \mathbb{R} > 0 $ is unknown. Further, let
$$ \| B x \|^2 < \frac{1}{\delta} \tag{2} \label{2} $$
be a bound on $Bx$ where $ x \in \mathbb{R}^n $. The goal is to maximize $\delta$ such that the bound in \eqref{2} is minimized while ensuring the inequality in \eqref{1} is satisfied.
LMIs are not something I have a ton of experience with and I have a couple of questions.
Question 1
If $ B^{\text{T}} B $ is invertible, then we can write (1) as;
$$ A (B^{\text{T}} B)^{-1} > \delta I_n \tag{3} \label{3} $$
can I select $$ \delta < \lambda_{\min}\big(A (B^{\text{T}} B)^{-1} \big) $$
to satisfy \eqref{3}? If so, is there a better way to select $ \delta $?
Question 2
Lets say $ B^{\text{T}} B $ is not invertible. Can I still solve for $ \delta $? I am completely lost on how to approach this case.