Análisis de mayorizaciones para distribuciones de probabilidad, matrices estocásticas de columna y sus preservadores lineales

Abstract

This work studies different bulge relations for matrices, with particular attention to strong bulge, on the sets of probability distributions and column stochastic matrices, and characterizes the linear operators that preserve them. These relations are induced by vector bulge, formally introduced by Hardy, Littlewood, and Pólya, which establishes a partial order among vectors by measuring their degree of dispersion. Its extension to the matrix case gives rise to the notions of strong, directional, and weak bulge. It is shown that the study of these relations can be reduced, without loss of generality, to the case of column stochastic matrices. With respect to the vector case, we recover Ando's characterization of linear bulge preservers on the set of probability distributions and establish that preserving bulge on 1^n or 0^n is equivalent to preserving it on all of R^n. For the matrix case, the main result of this work is the complete characterization of the linear operators Φ on Mn,m that preserve strong majorization on Ω_col{n,m}. This result extends and unifies the previous characterizations by Ando and Li-Poon, showing that restricting the matrix to column stochastic matrices enriches the structure of the linear preservers compared to the general case.