  In this article, we present a geometric theoretical analysis of semidefinite
feasibility problems (SDFPs). This is done by decomposing a SDFP into smaller
problems, in a way that preserves most feasibility properties of the original
problem. With this technique, we develop a detailed analysis of weakly
infeasible SDFPs to understand clearly and systematically how weak
infeasibility arises in semidefinite programming. In particular, we show that
for a weakly infeasible problem over $n\times n$ matrices, at most $n-1$
directions are required to approach the positive semidefinite cone. We also
present a discussion on feasibility certificates for SDFPs and related
complexity results.
