Unresolved intersection of positive semidefinite matrix classes

Determine whether there exist matrices that are simultaneously positive semidefinite of types 1 and 5 but neither positive semidefinite of type 2 nor positive semidefinite of type 3.

Background

The paper presents Venn diagrams describing relationships among five inequivalent notions of positive semidefiniteness over finite fields. The authors explicitly caution that not all regions and inclusions in the diagrams have been verified and that the figures are heuristic.

One unresolved region concerns the possible existence of matrices belonging to both the type-1 and type-5 classes while belonging to neither the type-2 nor type-3 classes.

References

For example, in \cref{fig:psd-venn}(A), we do not know if there are matrices which are both $\PSD_1$ and $\PSD_5$ but neither $\PSD_2$ nor $\PSD_3$.

— Positive definite, positive semidefinite and totally positive matrices over finite fields  (2608.17702 - Ayyer et al., 18 Aug 2026) in Remark following Figure 2, Section “Relating these definitions” of the positive semidefinite matrices section