  This paper is concerned with generalizations of the notion of principal
eigenvalue in the context of space-time periodic cooperative systems. When the
spatial domain is the whole space, the Krein-Rutman theorem cannot be applied
and this leads to more sophisticated constructions and to the notion of
generalized principal eigenvalues. These are not unique in general and we focus
on a one-parameter family corresponding to principal eigenfunctions that are
space-time periodic multiplicative perturbations of exponentials of the space
variable. Besides existence and uniqueness properties of such principal
eigenpairs, we also prove various dependence and optimization results
illustrating how known results in the scalar setting can, or cannot, be
extended to the vector setting. We especially prove an optimization property on
minimizers and maximizers among mutation operators valued in the set of
bistochastic matrices that is, to the best of our knowledge, new.
