Majorization via generalized Hessenberg matrices Original Research Article

In this paper we give a method of constructing generalized Hessenberg matrices from those of smaller orders, with a simple combinatorial justification. Making use of this construction, we also prove that, for real n-vectors x and y whose components are arranged in nonincreasing order, x is majorized by y if and only if there exists a doubly stochastic matrix A of order n with x=Ay whose support is permutation similar to a direct sum of generalized Hessenberg matrices.