14 problems

Let $X$ be an $N\times N$ matrix, let $z$ be an eigenvalue, and let ${\bf v}$ and ${\bf u}$ be corresponding right and left eigenvectors. Normalize the right eigenvector by…

Eigenvector coordinate conjecture. With high probability,

For each $k\geq 3$, let $\Delta_k$ be the maximum Chebyshev distance between the coordinatewise powers of the principal eigenvectors of a connected $k$-graph and its clique-shadow:…

Let $G$ be the random graph and let $\nu_1,\dots,\nu_n$ be the eigenvectors appearing in Theorem 1 of the paper, with $p$ as in that theorem. For a vector $v$, write $\|v\|_\infty$…

Eigenvector generation conjecture. For each $n\geq 1$, a generating set of $S_{\mathbb Q}\langle a,b\rangle_n$, consisting only of eigenvectors, may be obtained by applying all pos…

Expanding orthogonal similarity variant conjecture. For almost all matrices of dimension at most $7$, repeatedly applying this eigenvector construction and random orthogonal or uni…

Let $A$ be a matrix of dimension at most $7$, and repeatedly construct eigenvector matrices $X_i$ whose columns have unit $\mathbb{L}_2$ norm. A matrix property holds for almost al…

Let $X$ be a random matrix satisfying the assumptions of Theorem, and let $T$ be its Schur form. Define $Y=n|t_{12}|^2$, where $t_{12}$ is the relevant off-diagonal entry of $T$, a…

Let $A$ be a random banded matrix with independent entries and bandwidth $b$, and let $l$ denote its eigenvector localisation length. Localization-length conjecture. The eigenvecto…

Let $E$ be distributed according to the arcsine law, $U$ be uniform on $[0,1]$, and let $\mathcal{Z}$ be a standard two-sided Brownian motion started from $0$, independent of $E$ a…

Let $E$ be distributed according to the arcsine law, $U$ be uniform on $[0,1]$, and let $\mathcal{Z}$ be a standard two-sided Brownian motion started from $0$, independent of $E$ a…

Inverse Power Method conjecture. After $K$ iterations of the Inverse Power Method, the coordinate $X_1$ is evaluated with high relative accuracy; more specifically, its relative ac…

Let $G(n,p)$ be an Erdős–Rényi random graph. Assume … for some constant $\epsilon>0$. Let $v$ be a random unit vector uniformly distributed on the $(n-1)$-dimensional unit sphere,…

Nonvanishing conjecture. Almost surely, every eigenfunction $f$ of $G$ has empty zero set: