8 problems

Let ${\textrm{SO}}^\star(5)$ denote the variety of orthogonal matrices with the relevant zero pattern, and let a totally real linear slice mean a codimension-two linear section who…

Let $\Lambda_1\in\mathbb{R}^{m\times n}$ and $\Lambda_2\in\mathbb{R}^{n\times n}$ be diagonal matrices, with $\Lambda_2\succ0$ having distinct diagonal entries, and let…

For a positive integer $\ell$, an $\ell\times\ell$ orthogonal matrix $F\in O(\ell)$ is flat when all its entries have absolute value of order $\ell^{-1/2}$; in particular, it is lo…

Optimality conjecture. This lower bound is optimal on $\operatorname{O}(n)$, namely

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…

Let $O(d)$ denote the group of real $d\times d$ orthogonal matrices, and let $f$ be the objective function for the effective Givens approximation considered above. A local minimum…

The symmetric rational orthogonal zero-pattern conjecture. For any hermitian $X\in U(n)$ there exists a symmetric $Z\in O_n(\mathbb{Q})$ such that $\underline{X}=\underline{Z}$ and

Let $X$ be an $m\times n$ matrix over a field. Its zero-pattern (or support) is the $(0,1)$-matrix $\underline{X}$ defined by … Let $O(n)$ denote the real orthogonal $n\times n$ ma…