A Linear Algebraic Reconstruction of Majorization

Majorization is an important partial order on multisets. Since its development at the turn of the twentieth century by economists formalizing the notion of `exchange' among distributions, it has found applications ranging from stochastic scheduling to singular value theory. In this paper we reconstruct majorization as a partial order on vectors (ordered sequences) using linear algebra. We do this in two different ways, one following a recent extension of majorization to permit partial ordering among the indices of the sequences, and the other following a model of majorization as `exchangeability'. Majorization with respect to a partial order is shown to be a special case of majorization as exchangeability. In this special case, the resulting majorization order always defines a distributive lattice that is isomorphic to the standard real vector lattice. In classical majorization theory, the semigroup of doubly-stochastic matrices plays a crucial role, both in the definition of majorizat..