77 problems

Leading-block and eigenvalue conjecture. This leading principal diagonal block of $A^{-1}$ is a $\mathbf{Z}$-matrix. Moreover, $A$ has exactly one negative eigenvalue.

Let $k$ be a number field, let $M\in \mathrm{M}_n(\mathcal{O}_k)$, and suppose that the triangularization variety $X_M$ has a rational point, $X_M(k)\neq\varnothing$. An irreducibl…

The Fooling-Set-Submatrix problem takes integers $n,m$ and an $m\times m$ $0/1$-matrix $A$ as input, and asks whether $A$ contains a fooling-set submatrix of size $n$. NP-hardness…

Let $n\ge 3$, let $\mathbb{H}_n=\{(0,n,0,0),(0,n-2,0,2),(2,n-2,0,0)\}$, and let an $n\times n$ sign pattern ${\cal A}$ require $\mathbb{H}_n$ when … A sign pattern is irreducible i…

Quadratic bound conjecture. There is a constant $K$ such that every such set has a product of length smaller than $Kn^2$ with positive entries.

Let $A$ and $B$ be two hypomorphic matrices, meaning that $B=\Sigma(A)$ for a hypomorphism $\Sigma$ as in the surrounding reconstruction setting. Let $\lambda_i$ be a simple eigenv…

Let $S_n$ be the permutation group on $n$ letters and let $\mathbb Z_2^n$ denote the group of diagonal sign matrices. Twisted-product refinement. The group $G(A)$ in the spectral r…

Let $A$ be a real $n \times n$ symmetric matrix. For each $i$, write $A_i$ for the matrix obtained by deleting the $i$th row and column, and let $\operatorname{eigen}(A)$ denote th…

The Asymptotic Lower $r$-Permanent Conjecture. Under these hypotheses,

Extended local-permutation conjecture. Every element of $E(\mathfrak{t}_{\mathrm{loc}})$ is similar to a block-shift matrix via an element of $\Pi_{\mathrm{loc}}^{\mathrm{ex}}$.

Determinant and inverse conjecture. One has

Bernoulli-moment characterization conjecture. There is a family of functions $r(k,n,\nu):\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ such that a matrix $S$ is a distinguished matrix of a si…

Cecotti–Vafa spectrum conjecture. The Cecotti–Vafa construction can be made precise so that

Let $n\in\mathbb{N}-\{0\}$ and let $K$ be a field. Let $S$ be an affine subspace of $M(n\times n,K)$ such that every element of $S$ is nilpotent, and assume that each element of…

Let $M$ be a strictly upper triangular $(0,2)$-matrix, meaning that its entries are $0$ or $2$, and suppose that for each $k\geq 3$ there is at most one pair $(i,j)$ with $j-i=k$ a…

Let $n$ be a positive integer. A braid projection is a braid diagram without over/under-crossing information. Its CN matrix $N(B)$ is the zero-diagonal matrix whose $(i,j)$ entry c…

Let $n$ be a positive integer. For an $n$-braid diagram, its OU matrix $U(b)$ is the non-negative integer zero-diagonal matrix whose $(i,j)$ entry counts crossings between strands…

Let $n$ be a positive integer, and let $M$ be an $n\times n$ integer matrix. A zero-diagonal matrix is T0 if, whenever $1\leq i<j<k\leq n$, the conditions $M(i,j)=M(j,k)=0$ imply…

Characterization conjecture. An $n\times n$ matrix $M$ is the crossing matrix of some positive pure braid if and only if $M$ is a non-negative integer T0 symmetric matrix.

Ryser's conjecture. The permanent of $J^*$ is maximal in $\mathscr{U}(v,k)$.

Let $T$ be the $11\times 11$ matrix displayed in the paper, and let $\tilde{T}$ be any vertex multiplication of $T$. Vertex-multiplication conjecture. One should have … This conjec…

Singularity conjecture. The matrix $B_p(n-1)$ is singular if and only if

Multiplicity-matrix conjecture. The matrix $\mathrm{\bf N}_n$ is upper triangular. Furthermore, if $N_{u,v}$ is nonzero, then $u\succeq v$ and $\mathcal{h}(u)=\mathcal{h}(v)$. This…

Let $\mathcal{A}\subset\mathbb{R}$ be a non-empty finite set, and let $T(\mathcal{A})$ denote the quantity counting the relevant commuting pairs of $2\times 2$ matrices associated…

Let $\operatorname{Mat}^{n\times n}_q$ be the additive group of $n\times n$ matrices over $\mathbb F_q$. For integers $r_1,\ldots,r_k,t$ with $0\le r_i,t\le n$, let…