52 problems

Let $A_n$ be an $n\times n$ random matrix whose entries are independent and identically distributed random variables taking values in $\{\pm1\}$. Let $s_n(A_n)$ denote its smallest…

Cook's conjecture. The probability that a uniformly random element of $\mathcal{M}_{n,d}$ is singular should be exponentially small in $n$.

Von Neumann–Goldstine conjecture. The typical value of $s_{\min}(G)$ is of order $n^{-1/2}$, while the condition number $\kappa(G)$ is of order $n$.

Edelman's universality conjecture. The same limiting distribution should hold for other entry distributions, such as Bernoulli entries; in particular, for fixed $\varepsilon>0$,

Zhan's conjecture. For every $j=1,2,\ldots,n$, one should have

Rotationally symmetric non-holomorphic N-transform conjecture. The two $N$-transforms are related by

Weak Schatten estimate conjecture. One has

Let $H_1$ and $H_2$ be the operators described in the preceding theorem, and let $4e2_n$ denote the singular values of the relevant difference of semigroups, with spatial dimension…

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 $H_N$ be the unscaled histopolation matrices associated with the nonuniform-node families considered above, and suppose that the sequence $\{H_N\}_N$ has a GLT nature. Let…

Let $M$ be an $n\times n$ random matrix with real entries, and let $s_n(M)$ denote its smallest singular value, defined by … Here $\mathbb{S}^{n-1}:=\{x\in\mathbb{R}^n:\|x\|_2=1\}$…

Let $1<p<\infty$ and $1\leq k\leq n$. For a matrix $A$ and matrix subspace ${\mathcal M}$, let $R^{(p,k)}=A-B^{(p,k)}$ range over residuals of best approximations with respect to t…

Let ${\mathcal M}$ be a matrix subspace, let $A$ be a matrix, and let $Y_p$ be the unique best approximation to $A$ in ${\mathcal M}$ with respect to the $c_p$ norm for…

Let $\mathbb{F}$ be a field, and consider the result proved in the source for $\mathbb{F}=\mathbb{R}$ concerning a method based solely on singular vector spaces. Complex-field anal…

Improved-constant conjecture. The stated probability bound holds for the randomized subspace approximation setting under the hypotheses of the CDF theorem.

Let $A$ be an $m\times n$ real matrix of rank $r$, and let $S\in\mathbb{R}^{s\times m}$ be an oblivious sketching matrix. Write $\sigma_{\max}(A)$ and $\sigma_{\min}(A)$ for the la…

Let $A_n$ be a random symmetric matrix, and consider its singular values across the spectrum, including values near the origin and near the spectral edge. The paper establishes min…

Let $A$ be an $n\times n$ random matrix. In the symmetric-matrix setting, compare the joint least-singular-value estimates proved in the paper with the corresponding estimates for…

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 $z=(z_1,\ldots,z_n)$ be the eigenvalues and let $a=(a_1,\ldots,a_n)$ be the fixed squared singular values. Write $\Delta_n$ for the Vandermonde determinant, let…

Let $A$ be a random matrix with a doubly stochastic variance profile, meaning that its expected squared column and row norms satisfy … and … For a complex number $z$, write…

Let $\mathbb{H}_n$ be the $n$-dimensional truncation of the Hilbert matrix, and let $\sigma_i(\mathbb{H}_n)$ denote its $i$th singular value. Gerth's conjecture. For every fixed…

Let $A_n$ be a random symmetric matrix with independent subgaussian entries, and let $\sigma_{\min}(A_n)$ denote its least singular value. Suppose that a bound has the form … for a…

Let $H_n$ be the $n\times n$ Hilbert matrix, and let $\sigma_i(H_n)$ denote its $i$th singular value for a fixed index $i$. Limiting singular-value conjecture. Every singular value…