Statement. To ensure the rank of $\text{ddiag}(AQQ^T)-\sigma\Delta=n$, it is sufficient to require $\min_i(\text{diag}(AQQ^T))_i>\sigma\|\Delta\|$.

Note: $Q\in\mathbb{R}_{n\times 2}$, $\sigma$ is a scalar constant, $\Delta$ is $n\times n$ random matrix.

I don't know what this holds. And also, could someone point out which theory or which direction this statement belongs to, so that next time I can be more clear about where I can find this kind of techniques. Is it just basic linear algebra or something more advanced?