Linear preservers of circulant majorization on Rn

Let Mn,m be the set of all n×m real matrices. An n×n real matrix A is called circulant doubly stochastic if it is a convex combination of circulant permutation matrices. For x,y∈Rn, x is said to be circulant majorized by y (written as x≺cy), if there exists a circulant doubly stochastic matrix D such that x=Dy. In this paper, the concept of circulant majorization is investigated and then the linear preservers and strong linear preservers of this concept are characterized on Rn.