  Topological edge modes are excitations that are localized at the materials'
edges and yet are characterized by a topological invariant defined in the bulk.
Such bulk-edge correspondence has enabled the creation of robust electronic,
electromagnetic and mechanical transport properties across a wide range of
systems, from cold atoms to metamaterials, active matter and geophysical flows.
Recently, the advent of non-Hermitian topological systems---wherein energy is
not conserved---has sparked considerable theoretical advances. In particular,
novel topological phases that can only exist in non-Hermitian systems have been
introduced. However, whether such phases can be experimentally observed, and
what their properties are, have remained open questions. Here, we identify and
observe a novel form of bulk-edge correspondence for a particular non-Hermitian
topological phase. We find that a change in the bulk non-Hermitian topological
invariant leads to a change of topological edge mode localisation together with
peculiar purely non-Hermitian properties. Using a quantum-to-classical analogy,
we create a mechanical metamaterial with non-reciprocal interactions, in which
we observe experimentally the predicted bulk-edge correspondence, demonstrating
its robustness. Our results open new avenues for the field of non-Hermitian
topology and for manipulating waves in unprecedented fashions.
