3 problems

Let $\mathcal{A}$ be a finite-dimensional $\ast$-algebra over $\mathbb{R}$, and let $H\in\mathcal{A}$ be self-adjoint, meaning $H^{\ast}=H$. Let $k\in\mathbb{N}$ be maximal among t…

Let $(\mathcal A,*)$ be an associative, unital, finite-dimensional $*$-algebra over $\mathbb R$. The spectral monoid $\overline{\operatorname{Her}}(\mathcal A,*)$ is the monoid of…

DODD existence conjecture. The $d\times d$ 0-inflation of any matrix $\mathbf{X}\in\mathbb{R}^{m\times n}$ admits a DODD for some sufficiently large $d\geq m,n$. Thus every such ma…