  We develop a general theory of flat-band ferromagnetism in the SU($N$)
Fermi-Hubbard model, which describes the behavior of $N$-component fermions
with SU($N$) symmetric interactions. We focus on the case where the
single-particle spectrum has a flat band and establish a necessary and
sufficient condition for the SU($N$) Hubbard model to exhibit ferromagnetism
when the number of particles is the same as the degeneracy. We show that the
occurrence of ferromagnetism is equivalent to the irreducibility of the
projection matrix onto the space of single-particle ground states. We also
demonstrate that this result can be exploited to establish a rigorous result
for the ferromagnetic SU($N$) Kondo lattice model with a flat band.
Specifically, we prove that when the SU($N$) Hubbard model is ferromagnetic,
the ferromagnetic SU($N$) Kondo lattice model with the same hopping matrix also
exhibits SU($N$) ferromagnetism.
