Bad Projections of the PSD Cone

TL;DR

This paper investigates the geometric properties of linear maps that produce non-closed images of the positive semidefinite cone, characterizing their structure and developing algebraic tools for explicit analysis.

Contribution

It characterizes the set of 'bad' projections of the PSD cone and explores their algebraic and geometric properties using convex algebraic geometry.

Findings

The set of non-closed projections forms a hypersurface in the Grassmannian.

Components are coisotropic hypersurfaces of symmetric determinantal varieties.

Provides explicit computational methods for analyzing these projections.

Abstract

The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.