12 problems

Let $n\geq 3$. A linear subspace of $\operatorname{M}_n(\mathbb{F}_2)$ is $\overline{2}$-spec when every matrix in it has characteristic polynomial of one of the allowed forms over…

Let $n\geq 2$. A linear subspace of $\operatorname{M}_n(\mathbb{F}_2)$ is $\overline{1}^\star$-spec when every matrix in it has minimal polynomial of the form $t^a(t-1)^b$. Dimensi…

Let $n\geq 2$. Consider $\overline{1}^\star$-spec linear subspaces of $\operatorname{M}_n(\mathbb{F}_2)$ having the greatest possible dimension, conjecturally $\dbinom{n+1}{2}$. A…

Let $n\geq 2$ and let $\mathbb{F}$ be a field. Write $t_n(\mathbb{F})$ for the greatest possible dimension of a weakly triangularizable linear subspace of…

Let $r=2s$, let $\bbF$ be the underlying field, and let $\mathcal{V}$ be a trivial-spectrum linear subspace of $\operatorname{M}_s(\bbF)$ with dimension $\frac{s(s-1)}{2}$. Conside…

Fix positive integers $n$ and $k<n$, and let $V$ be a linear subspace of $M_n(\mathbb{C})$ such that every matrix in $V$ has at most $k$ distinct eigenvalues. Loewy and Radwan prov…

Let $V$ be a subspace of $M_n(\mathbb{C})$, and let $k<n$. Assume that … for all $A,B\in V$. Define $V_k$ to be the space of block matrices … where $r_k$ and $s_k$ are, in either o…

Inherited correspondence conjecture. The maximum number of edges in a subgraph $H$ of $G$ in which every collection of disjoint cycles covers at most $k$ vertices equals the larges…

Let $\mathbb{K}$ be a field of characteristic $2$ with more than $2$ elements. Let $\mathcal{V}$ be a $\overline{2}$-spec subspace of $\operatorname{M}_n(\mathbb{K})$, with…

Let $\mathbb{K}$ be a field of characteristic $2$ with more than $2$ elements. Let $\mathcal{V}$ be a $\overline{1}^\star$-spec subspace of $\operatorname{M}_n(\mathbb{K})$ with ……

Let $\mathbb{K}$ be a field of characteristic zero, let $n>k$, and let $\mathcal{V}$ be a $\overline{k}$-spec subspace of $\operatorname{M}_n(\mathbb{K})$. LoewyRadwan's conjecture…

Let $V$ be a $1$-spec subspace of $\operatorname{M}_n(\mathbb{K})$ such that … Assume $n>4$. A linear subspace $W$ of $\operatorname{M}_n(\mathbb{K})$ has a trivial spectrum if no…