  The uniqueness of the Perron vector of a nonnegative block matrix associated
to a multiplex network is discussed. The conclusions come from the
relationships between the irreducibility of some nonnegative block matrix
associated to a multiplex network and the irreducibility of the corresponding
matrices to each layer as well as the irreducibility of the adjacency matrix of
the projection network. In addition the computation of that Perron vector in
terms of the Perron vectors of the blocks is also addressed. Finally we present
the precise relations that allow to express the Perron eigenvector of the
multiplex network in terms of the Perron eigenvectors of its layers.
