On the positivity of matrix-vector products

Abstract In this paper we examine the positivity of Rv where R∈ R N×N , v∈ R N , v ⩾0 with R = r ( τA ), r is a given (rational) function, A∈ R N×N and τ ∈(0,∞). Here we mean by positivity the ordering w.r.t. an arbitrary order cone, which includes the classical entrywise positivity of vectors. Since the requirement R ⩾0 leads to very severe restrictions on r and τ we construct a positive cone P = P (A) and determine τ ∗ =τ ∗ (r, P ) such that r(τA) P ⊂ P for all τ∈[0,τ ∗ ] . Finally we give an example arising from applications to partial differential equations where our results explain actual computations much better than the general theory on R ⩾0.