36 problems

Let $\mathbf{u}_1,\ldots,\mathbf{u}_m\in\mathbb{C}^{d}$ satisfy $\|\mathbf{u}_i\|\leq 1$ for all $i$, and suppose there are universal constants $\eta\geq 2$ and $\theta>0$ such tha…

Let $A,B e 0$ be Hermitian positive semidefinite matrices of the same size, and let $Acirc B$ denote their Hadamard product. Chollet's conjecture. … The conjecture is a permanent a…

Bapat–Sunder's conjecture. The largest eigenvalue of $F_A$ should satisfy

Let $\mathcal{H}_n$ be the set of $n\times n$ positive semidefinite Hermitian matrices. For matrices $A=[a_{i,j}]$ and $B=[b_{i,j}]$ in $\mathcal{H}_n$, define the Hadamard product…

Positive semidefinite Graph Complement Conjecture. For any graph $G$,

Positive semidefiniteness conjecture. The matrix $M_r$ is positive semidefinite for all $r$.

Let $Q^{(i)}=[Q^{(i)}_{(jk)}]_{j,k=1}^2$ be arbitrary positive semidefinite $2\times2$ block matrices, and let $q^{(i)}_k$ be arbitrary non-negative numbers. For a matrix $A$, writ…

Bapat's conjecture. The minimum over doubly stochastic tuples is

Hillar's commuting-minimizers conjecture. Such matrices $A$ and $B$ commute.

Rank-$r$ projection sparsification conjecture. There exists a constant $C_r$ such that one can choose indices $i_1,\dots,i_N\in[m]$ with

Let $A_1,\dots,A_n\in\mathcal{M}_d$ be positive semidefinite, and suppose that nonnegative coefficients $\lambda_1,\dots,\lambda_n$ summing to one satisfy … For…

Matrix-analysis conjecture. Under these hypotheses,

Let $\mathcal{H}_n$ be the set of $n\times n$ positive semidefinite Hermitian matrices. For $A=[a_{i,j}]\in\mathcal{H}_n$, let $A(i,j)$ be obtained by deleting row $i$ and column…

Let $\mathcal{H}_n$ be the set of $n\times n$ positive semidefinite Hermitian matrices. For matrices $A=[a_{i,j}]$ and $B=[b_{i,j}]$ in $\mathcal{H}_n$, define…

Let $A\in\mathcal{M}_n^{+}$ be entrywise non-negative and have factor width at most $k$, and let $s>0$. The Hadamard power $A^{\odot s}$ is defined entrywise. Factor-width conjectu…

Lieb–Seiringer conjecture. This coefficient is non-negative.

Border-rank gap conjecture. There is a nonnegative tensor $T$ such that

Let $P(\lambda)=\lambda(J_1+R_1)+(J_2+R_2)$ be a posH pencil, and let $\lambda J_1+J_2$ be its underlying skew-Hermitian pencil. Right-half-plane eigenvalue localization conjecture…

Pairwise-max positive-semidefiniteness conjecture. The matrix $R(\overrightarrow{r}_{pm})$ is always positive semidefinite; equivalently, $\overrightarrow{r}_{pm}$ is a valid choic…

Square-root-degree conjecture. Using arguments of Newman and Rivlin, $N=O(\sqrt{q})$ should be sufficient to accurately approximate $\mathbf{A}^p$.

Novak's conjecture. The matrix

The tree cover bound. For every graph $G$,

Let $P$ be a lattice polytope containing a lattice point in its interior, and let $h^2_i(P)$ denote the coefficient matrices of its $h^2$-tensor polynomial. Hibi-type conjecture. F…

Let $P$ be a lattice polytope in $\mathbb{R}^d$, and let $h^2(P)$ denote its $h^2$-tensor polynomial, with $d\geq 1$. Positive-semidefiniteness conjecture. The coefficients of…

Let $A$ and $B$ be $n\times n$ positive semidefinite matrices. Let $\lambda(X)$ denote the vector of eigenvalues of a Hermitian matrix $X$, arranged in decreasing order, and let…