  Multiplex networks are networks composed of multiple layers such that the
number of nodes in all layers is the same and the adjacency matrices between
the layers are diagonal. We consider the special class of multiplex networks
where the adjacency matrices for each layer are commuting pairwise. For such
networks, we derive the relation between the spectrum of the multiplex network
and the eigenvalues of the individual layers. As an application, we propose a
generalized master stability approach that allows for a simplified,
low-dimensional description of the stability of synchronized solutions in
multiplex networks. We illustrate our result with a duplex network of
FitzHugh-Nagumo oscillators. In particular, we show how interlayer interaction
can lead to stabilization or destabilization of the synchronous state. Finally,
we give explicit conditions for the stability of synchronous solutions in
duplex networks of linear diffusive systems.
