229 problems

Lee's conjecture. The optimal constant for the Schatten $2$-norm is

Strict growth conjecture. For every positive integer $n$, one has

Kadison–Singer conjecture. There is a partition $X_1,\ldots,X_r$ of $[m]$ such that

Let $f:\mathbb{C}\to\mathbb{C}$ act entrywise on complex Hermitian matrices, and suppose that the functions $z\mapsto z^j\overline{z}^k$ for $j,k\geqslant 0$ preserve positive semi…

Let $S\in\mathcal{M}_d(\mathbb{C})^+$ and let $\mathbf{a}=(a_i)_{i\in\mathbb{I}_k}\in(\mathbb{R}_{>0}^k)^\downarrow$. Define … and, for a strictly convex unitarily invariant norm…

Let $T$ be an $N\times N$ normal matrix, and let $k\geq 1$ be a fixed positive integer. Write $\Lambda_k(T)$ for the rank-$k$ numerical range of $T$, and let $\Gamma$ range over al…

Let $M:\mathbb{R}^{d\times d}\to\boxed{?}$ be a linear map, and let … be the Schatten-1 ball. Volume-ratio conjecture. Is it true that … A positive answer would give an…

Let $A_1$ and $A_2$ be $n\times n$ Hermitian matrices, and define their characteristic polynomial by … Suppose that $P_A$ has a repeated factor in the polynomial ring…

Daubechies–Lagarias conjecture. The generalized spectral radius and the joint spectral radius are equal for any finite set of matrices.

Choi–Kribs–Życzkowski's convexity conjecture. Higher-rank numerical ranges $\Lambda_k(A)$ are always convex. This conjecture was proved by Woerdeman in 2008, so the claim is solved…

Let $X\in\mathbb{M}_n$. Say that $X$ has the universal block-matrix inequality if, for every positive block matrix with $X$ as its off-diagonal block, … Essentially Hermitian conje…

Let $A$ be an $n$-by-$n$ nilpotent matrix, and let $W(A)$ denote its numerical range. A flat portion is a line segment contained in the boundary of $W(A)$. Gau–Wu conjecture. The b…

Audenaert–Datta conjecture. If $1\leq p\leq 2$, $-1\leq q<0$, and $(p,q)\ne(1,-1)$, then $\Psi_{p,q,1/(p+q)}(A,B)$ is jointly convex in $(A,B)$. This conjecture concerns the joint…

BMV conjecture. The function $f_{A,B}$ is exponentially convex.

Let $R$ be a positive semidefinite matrix on a bipartite tensor product, with partial traces denoted by ${\rm Tr}_1$ and ${\rm Tr}_2$. For a tripartite positive semidefinite matrix…

Entrywise nonnegativity conjecture. This equality actually implies that

Let a density matrix be a positive definite matrix of trace $1$, and let its eigenvalues be the corresponding positive numbers summing to $1$. The scalar curvature is computed for…

Let $A$ be a matrix of arbitrary finite size, and consider the essential fault points associated with the boundary of its pseudospectrum. Finiteness conjecture. The number of essen…

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…

Let $D_1,D_2,D_3$ be positive definite matrices, and let $M$ be the matrix whose entries are the coefficients of $y$ and $z$ in … A minor means the determinant of any finite square…

Let $f(x,y)\in\mathbf{P}_2$ and $g\in\mathbf{P}$. A polynomial satisfies derivative-substitution when all its derivatives with respect to $y$, evaluated at every real $y$-value, ar…

Let $A_i,B_i$ be positive definite matrices, let $g_i\in\mathbf{P}^{\mathrm{pos}}$, and let $f(x,y_1,\dots,y_e)\in\mathbf{P}_{1+e}$. The strong BMV conjecture. The determinant … ha…

Let $A$ and $B$ be positive definite matrices, and let $n$ be a positive integer. The BMV conjecture. The polynomial in $t$ … has all positive coefficients. The source presents thi…

Positivity conjecture. For every $k=0,1,\dots,r$, the polynomial $S_k((A+tB)^m)$ has only nonnegative coefficients.

Let $A$ be a Hermitian matrix, and let $\mathcal{X}$ and $\mathcal{Y}$ be subspaces of the same dimension, with orthonormal basis matrices $X$ and $Y$, respectively. Let…