11 problems

Schatten norm conjecture for r-partite matrices. If $A$ is $r$-partite, then

Let $\mathcal O(r)$ be the real square orthogonal group, and define … A signed permutation matrix is an orthogonal matrix with exactly one nonzero entry, equal to $1$ or $-1$, in e…

Let $A$ be an $n\times n$ non-negative, non-singular matrix. An S-matrix is a matrix in $\{0,1\}^{n\times n}$ satisfying … where $\boldsymbol{1}$ is the all-ones vector. Harwitz–Sl…

Let $U\in\mathbb{C}^{n\times n}$ be a scaled upper-triangular factor of an incomplete LU factorization, written as $U=I+U_s$, where $I$ is the identity matrix and $U_s$ is strictly…

Let $X$ be either ${\mathcal{M}}_n({\mathbb{F}})$ or ${\mathcal{H}}_n({\mathbb{F}})$, let $\|\cdot\|$ be an admissible norm on $X$, and write $k=\dim X$. For a placement $p\in X^V$…

Multichain exclusion conjecture. The condition $\lVert (R-P)D_P\rVert<1$ rules out the case that $R$ is a multichain.

Spectral p-norm equality conjecture. For every integer $r\geq 3$, there is a $p_0(r)\in(1,2)$ such that

Let $\boldsymbol{S}$ be a finite family of complex matrices. It is non-defective if the semigroup generated by the set $\rho(\boldsymbol{S})^{-1}\boldsymbol{S}$ is bounded. Suppose…

Let $r,d \in \mathbb{N}$, let $\mathcal{I}_r(\mathbb{K}^d)$ be the ambient class of $r$-tuples under consideration, and let $\Omega_r^n$ denote the words of length $n$. For…

Let $\|Y\|_{CL}$ and $\|Y\|_{NC}$ be the Carlen–Lieb and non-commutative norms, respectively, on matrices $Y\in {\cal M}_n\otimes {\cal M}_m$, with parameters $1<p\le2$ and $q\ge1$…

Let $\|Y\|_{CL}$ and $\|Y\|_{NC}$ denote the two non-commutative versions of the classical $L^q(L^p)$ norm on the matrix algebra ${\cal M}_n \otimes {\cal M}_m$, defined respective…