  We consider a model for linear transient price impact for multiple assets
that takes cross-asset impact into account. Our main goal is to single out
properties that need to be imposed on the decay kernel so that the model admits
well-behaved optimal trade execution strategies. We first show that the
existence of such strategies is guaranteed by assuming that the decay kernel
corresponds to a matrix-valued positive definite function. An example
illustrates, however, that positive definiteness alone does not guarantee that
optimal strategies are well-behaved. Building on previous results from the
one-dimensional case, we investigate a class of nonincreasing, nonnegative and
convex decay kernels with values in the symmetric $K\times K$ matrices. We show
that these decay kernels are always positive definite and characterize when
they are even strictly positive definite, a result that may be of independent
interest. Optimal strategies for kernels from this class are well-behaved when
one requires that the decay kernel is also commuting. We show how such decay
kernels can be constructed by means of matrix functions and provide a number of
examples. In particular we completely solve the case of matrix exponential
decay.
