Consider the matrix $M = \begin{bmatrix} A & A B \end{bmatrix} \in R^{n \times (n+m)}$, with $A \in R^{n\times n}$, $B \in R^{n \times m}$, $m < n$, $m > 1$, $A$ symmetric positive definite.
I'm interested in the finding an (good as possible) upper bound for the following expression $$ \sup \{ 1 / \sigma_{min}(C) ~:~ R^{n \times n}\ni C = \begin{bmatrix} A & AB \end{bmatrix}_I, C ~\text{invertible} \}, $$ where the supremum is taken over all column selections $[.]_I$ of $M$ such that the resulting square matrix $C$ is invertible.
Basically I'm interested in the smallest singular value of all invertible $n \times n$ sub-matrices of $M$.
Right now, I have no idea what are even the right tools to attack this problem. I'd be even interested in some numerical way to find an upper bound for this number for values of $n \approx 500$, $m \approx 200$ but an exhaustive search quickly becomes completely intractable.
