110 problems

For every $n,m\ge 1$, every matrix $A\in M_n(\mathbb{C})$, and every matrix-valued polynomial $P(z)=\sum_{k=0}^d P_k z^k$ with $P_k\in M_m(\mathbb{C})$, one has…

Schur stability conjecture. The operators $P(t)$ and $R(t)$ are stable, meaning that

Let $A,B\in\mathbb{H}_n$ be Hermitian matrices, and let $H$ and $R_k$ denote the operators defined in the paper from the coefficients of $e^{(A+B)t}$ and $e^{At}e^{Bt}$, respective…

Let $\Lambda_1\in\mathbb{R}^{m\times n}$ and $\Lambda_2\in\mathbb{R}^{n\times n}$ be diagonal matrices, with $\Lambda_2\succ0$ having distinct diagonal entries, and let…

Let $\mathcal{M}$ be a semifinite von Neumann algebra endowed with a faithful normal semifinite trace $\tau$, and let $\mathcal{P}$ be the positive cone of the space of all $\tau$-…

Matrix-analysis conjecture. Under these hypotheses,

Near-isometric duality conjecture. For every $n\times n$ matrix $A$ with $n>2$,

Foci conjecture. If $\operatorname{cl}(W(T))$ is an elliptical disk, then its foci are contained in $\{0,1,-1\}$.

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…

For every integer $n\ge 1$ and every real symmetric positive definite matrix $G\in\mathbb{R}^{n\times n}$, the matrix $\left(G\circ G^{-1}\right)^{-1}$ is entrywise nonnegative, wh…

For every unitarily invariant matrix norm $\|\cdot\|_{\mathrm{ui}}$, does there exist a constant $c>0$, independent of $n$ and of $G\in\mathbb{C}^{n\times n}$, such that … Here…

Determine the least constant $c\in\mathbb{R}$ such that, for every $n\in\mathbb{N}$ and all contractions $A,B,C\in M_n(\mathbb{C})$, one has $|A+B+C|\leq cI_n+|A|+|B|+|C|$, where…

Given matrices $A$, $B$, and $C$, determine conditions under which the solution $X$ of the Sylvester equation $AX-XB=C$ is well-conditioned, rather than merely existing uniquely or…

For every integer $n\ge 1$, every matrix $A\in\mathbb{C}^{n\times n}$, and every polynomial $p\in\mathbb{C}[z]$, one has $\|p(A)\|_2\le 2\sup_{z\in W(A)}|p(z)|$, where…

For every $n\ge 1$, every unital stochastic map $\Phi:M_n\to M_n$, and every normalized monotone Riemannian metric specified by an operator-monotone function $\kappa$, the Riemanni…

For $m\ge 2$ and $1<p<\infty$, define the all-dimensional best constant $c_p(m)$ by … The Tang–Zhang conjecture asserts that $c_p(m)$ equals the explicit closed-form expression pro…

For Hermitian matrices $A_{1},A_{2}\in M_{n}(\mathbb{C})$, define $p(z_{0},z_{1},z_{2})=\det\!\left(z_{0}I+z_{1}A_{1}+z_{2}A_{2}\right)$. Kippenhahn's conjecture asserts that if th…

Direct-summand conjecture. If

Rump's 100 Euro Conjecture. There exists $x\in\mathbb{R}^n\setminus\{0\}$ such that

Let $A$ and $B$ be Hermitian $n\times n$ matrices, with $B$ positive semidefinite. Let $\operatorname{Tr}$ denote the trace. Bessis–Moussa–Villani conjecture. The function … is the…

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 $A$ and $B$ be real positive definite matrices, and let a symmetric word equation be an equation obtained by evaluating a symmetric word in $A$ and $B$ at a prescribed matrix.…

Let $\mathcal{N}(\mathbb{C}^n)$ denote the set of nilpotent operators on $\mathbb{C}^n$, and let $\delta_n$ be the distance from the set of non-zero projections in…

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