For every ${A\in\mathcal{M}_n(\mathbb{C})}$ let us define

$s(A):=\min_{\Vert x\Vert_2=1}\Vert Ax\Vert_2 \quad\text{and}\quad \Vert A\Vert:=\max_{\Vert x\Vert_2=1}\Vert Ax\Vert_2.$

Let ${X}$ be a random matrix in ${\mathcal{M}_n(\mathbb{C})}$ with i.i.d. entries of mean ${m:=\mathbb{E}(X_{11})}$ and unit variance. Fix ${0<s_-\leq s_+<\infty}$ and let ${A_1,\ldots,A_n}$ be invertible deterministic matrices in ${\mathcal{M}_n(\mathbb{C})}$ s.t.

$s_- \leq \min_{1\leq k\leq n}s(A_k) \leq \max_{1\leq k\leq n}\Vert A_k\Vert\leq s_+.$

Let ${R_1,\ldots,R_n}$ be the rows of ${X}$ and ${Y}$ the random matrix with rows ${R_1A_1,\ldots,R_nA_n}$ .

Conjecture (RV). If ${X_{11}}$ is sub-Gaussian, i.e. there exists ${c_0}$ such that for every ${t\geq0}$,

$\mathbb{P}(|X_{11}-m|>t)\leq 2 e^{-c_0t^2}$

then there exists ${C>0}$ and ${c\in(0,1)}$ depending (polynomially) only on ${m}$, ${c_0}$, ${s_{\pm}}$, such that for large enough ${n}$ and every ${\varepsilon\geq0}$,

$\mathbb{P}(s(Y)\leq \varepsilon) \leq C\varepsilon+c^n.$

Conjecture (TV). For every ${a>0}$ there exists ${b>0}$ depending only on ${a,c,m,s_{\pm}}$, such that for every deterministic matrix ${A\in\mathcal{M}_n(\mathbb{C})}$ with ${\Vert A\Vert=O(n^c)}$ and large enough ${n}$,

$\mathbb{P}(s(Y+A)\leq n^{-b}) \leq n^{-a}.$

These conjectures involve a transformation of ${X}$, which leaves invariant the results of Adamczak et al on the smallest singular values of random matrices with i.i.d. centered log-concave rows.

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