16 problems

Vu's distinct singular values conjecture. With probability $1-o(1)$, all the singular values of $M_n^{sym}$ are distinct.

Let $F=[g_1,g_2,\dots,g_{n+k}]$ be an $n\times(n+k)$ matrix, and define … Assume that $\|g_j\|\leq1$ for every $j$. Polynomial upper-bound conjecture. There exists a constant…

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…

Fix an integer $d\geq 2$ and a constant $C>0$. Let $r(k),n_k\in\mathbb{N}$ satisfy … For each $k$, let $M^{(k)}\in\{-1,0,1\}^{k\times n_k}$ be a deterministic matrix whose every co…

Let $M_n^{sym}$ be a symmetric random matrix whose entries are independent Rademacher variables. Vu's distinct singular values conjecture. With probability $1-o(1)$, all the singul…

Let $X$ be an $N\times N$ real Ginibre matrix, let $A$ be a deterministic matrix, let $\gamma>0$, and define the Hermitian imaginary part by … Write $\sigma_1(\gamma X+A)$ for the…

Lemos–Soares conjecture. If $A,B\geq0$, $r,s\in\mathbb{R}$, and $0\leq t\leq1$, then

Let $\sigma_{1,n}$ denote the largest singular value of the matrix associated with index $n$. Largest-singular-value bound. … This bound is suggested by the numerical homotopy expe…

Let $\boldsymbol{M}_n$ be a real random matrix satisfying the paper's Assumption with parameter $K>0$, possibly with some moment assumptions on its entries. For a deterministic mat…

Malkoun's singular-value conjecture. For every $\mathbf{x}\in X$,

Let $\xi$ be a mean-zero, variance-at-least-$1$, subgaussian random variable, and let $N_n$ be an $n\times n$ random matrix with iid entries distributed as $\xi$. The least singula…

Non-Gaussian hard-edge universality conjecture. Under assumptions (C1)--(C4), the conclusion of Theorem (hardlimits) should still hold, with $\kappa=0$. This conjecture extends the…

Consider the real counterpart of the joint probability density function in the paper's equation (matrixpdf), with the same parameters and scalings as in Theorem (hardlimits). Real…

Let $\alpha_{\mathbb{R}}(d)$ and $\alpha_{\mathbb{C}}(d)$ be the average singular value of a $d\times d$ matrix with random i.i.d. real-valued and complex-valued entries, respectiv…

Let $A,B\in\mathbb{R}^{m\times n}$ be given, and let $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be a concave function satisfying $f(0)=0$. For $k\in\{1,\ldots,\min\{m,n\}\}$, let…

Let $X,Y\in\mathbb{M}_{n,m}(\mathbb{C})$, and let $\sigma_i(X)$ denote the singular values in non-increasing order. Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be concave with $f(0)=0$. Mi…