105 problems

Let $A$ be a positive semidefinite Hermitian matrix, let $\lambda\vdash n$ be a partition, let $\chi^{\lambda}$ be the irreducible character of $\mathfrak{S}_n$ indexed by…

Böttcher–Wenzel conjecture. The commutator satisfies

Let $A$ and $B$ be non-negative $N\times N$ matrices. Write $A\circ B$ for their Hadamard product, $AB$ for their usual matrix product, and let $\rho$ denote the spectral radius. Z…

Let $n$ be a positive integer, $A_1,dots, A_n$ be symmetric positive semidefinite matrices, and let $\lVert\,\cdot\,\rVert$ be the spectral norm. For $m\leq n$, compare the average…

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Lu–Wenzel eigenvalue conje…

Let $\mathbb{K}=\mathbb{R}$ and let $B_1,\ldots,B_m\in M(n,\mathbb{K})$, where $M(n,\mathbb{K})$ is the space of $n\times n$ matrices over $\mathbb{K}$. Let $[A,B]=AB-BA$ and let…

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

Caputo–Carlen–Lieb–Loss conjecture. The maximum of the permanent of $A$ is attained either by $I$, with permanent $1$, or by the matrix $n^{-1/p}J$, with permanent $n!/n^{n/p}$. Eq…

Let $A,B\in\mathcal S_n$, and let $\lVert\!\cdot\!\rVert$ be a unitarily invariant norm on matrices. The notation $|AB|$ denotes $(AB)^*AB$ raised to the power $1/2$, so…

Newton's inequalities. For every M-matrix, inverse M-matrix, or matrix similar to one of these, the normalized characteristic-polynomial coefficients satisfy

Let $\rho$ be a quantum statistical model with a full parameter dimension $d^2-1$, and let $\rho'_i$ denote its parameter derivatives. Let $\boldsymbol{\mu}$ be the vector of symme…

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

Let $A$ and $B$ be complex $n\times n$ matrices. Write the singular values of $A$ as $\sigma_1\geq\dots\geq\sigma_n\geq0$, and define … For $1\leq p<\infty$, let…

Let $A$ be a complex $n\times n$ matrix with singular values … Let $\Sigma_\uparrow(A)=\operatorname{diag}(\sigma_1,\dots,\sigma_n)$ and…

Let $A$ and $B$ be complex $n\times n$ matrices, and for $1\leq p<\infty$ define $\|A\|_p=(\operatorname{Tr}|A|^p)^{1/p}$, where $|A|=(A^*A)^{1/2}$. The matrix Hanner inequality is…

Yun–Sra–Jadbabaie conjecture. There exists a constant $\eta_{n,K}\in(0,1]$ such that, whenever $(1-\eta_{n,K})I\preceq A_i\preceq I$ for every $i$, one has

Lin's conjecture. There exists a unitary matrix $U\in M_n$ such that

Pinching refinement. For all $n,m\ge0$,

Let $M$ be a matrix with non-negative real entries, let $t$ be an odd positive integer, and let $(x_1,y_1),\ldots,(x_t,y_t)$ be distinct entries of $M$ satisfying $M_{x_i y_i}=0$ f…

Weak Loewner spectral-dominance conjecture. The matrices $L$ and $D$ satisfy

Weak Loewner dominance conjecture. The connection Laplacian $L$ always dominates both $D$ and $g=L^{-1}$ in the weak Loewner sense; in particular, the proposed comparison with $D$…

Le's conjecture. Then $\operatorname{diag}(A)\in\Gamma(P)$ and

Let $M$ be any $n\times n$ symmetric matrix with entries satisfying $0\leq M_{ij}\leq1$, and let $\mathbf{1}$ be the $n$-dimensional all-ones vector. Matrix power inequality conjec…

Let $Z$ be an $n\times n$ positive semidefinite similarity matrix as above, and let $\mathbf{1}$ denote the $n$-dimensional all-ones vector. Equivalent bounds conjecture. For $q>1$…

Let $Z$ be an $n\times n$ positive semidefinite similarity matrix with $n>0$, symmetric entries in $[0,1]$, and diagonal entries equal to $1$. On the uniform distribution, write LC…