3 problems

Uniqueness conjecture. There is a unique matrix $Y\in K_n$ such that the smallest eigenvalue of $YY^T$ is $c_n$. Equivalently, if the smallest eigenvalue of $YY^T$ equals $c_n$, th…

Ilmonen–Haukkanen–Merikoski conjecture. The number $c_n$ is equal to the smallest eigenvalue of $Y_0Y_0^T$.

Let $\{x_i\}_{i=1}^{\infty}$ be the arithmetic progression described above, let $S_n=\{x_1,x_2,\ldots,x_n\}$, and let $\lambda_n^{(1)}\leq\cdots\leq\lambda_n^{(n)}$ be the eigenval…