4 problems

Let $\widetilde{A}$ be an $n\times n$ matrix satisfying $\|\widetilde{A}\|_2\leq\sqrt{n}$, and let $A$ be a Gaussian perturbation of $\widetilde{A}$ with variance $\sigma^2\leq1$.…

Let $\widetilde{A}$ be an $n\times n$ matrix with $\|\widetilde{A}\|_2\leq 1$, and let $A$ be a Gaussian perturbation of $\widetilde{A}$ with variance $\sigma^2\leq1$. Let $U$ be t…

Let $\widetilde{A}$ be an arbitrary square matrix in $\mathbb{R}^{n\times n}$, and let $A$ be a Gaussian perturbation of $\widetilde{A}$ with variance $\sigma^2$. Smallest singular…

Let $A$ be an arbitrary $n \times n$ matrix, and let $M_n$ be a random $n \times n$ matrix. For an invertible matrix $M$, define its condition number by … and set…