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

**Note:** $Q\in\mathbb{R}_{n\times 2}$, $\sigma$ is a scalar constant, $\Delta$ is $n\times n$ random matrix. The operator $\operatorname{ddiag}$ sets all off-diagonal entries of a matrix to zero. The operator $\operatorname{diag}$ takes the diagonal entries of a matrix.

I don't know why 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?