  We prove that Nagaoka's theorem, that the large-$U$ Hubbard model with
exactly one hole is ferromagnetic, holds for any balanced Hamiltonian. Simply
put, if a positive bond encodes friendship and a negative bond encodes enmity,
then balance implies when that enemy of one's enemy is one's friend. We argue
that, in itinerant electron systems, a balanced Hamiltonian, rather than
bipartite lattice, defines an unfrustrated system. The proof is valid for
multi-orbital models with arbitrary two-orbital interactions provided that no
exchange interactions are antiferromagnetic: a class of models including the
Kanomori Hamiltonian.
