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Majorizations for probability distributions, column stochastic matrices and their linear preservers

Pavel Shteyner

TL;DR

The paper addresses linear preservers of majorization for probability distributions and column stochastic matrices, establishing reductions to column-stochastic settings and solving a prerequisite zero-sum vector problem. It proves that preserving majorization on probability distributions is equivalent to preserving vector majorization, via reductions to $(0,1)$-vectors, and then delivers a complete characterization of linear preservers of strong majorization for column stochastic matrices. A key technical contribution is the description of preservers on the zero-sum space, showing they must have the form $[oldsymbol{ ext{φ}}] = v e^t + λ P$, which links to Ando-type preservers and clarifies the structure of linear preservers in this setting. The main result unifies Li–Poon and Ando-type operators within a single structural form, providing a richer preservers theory for matrix majorization in the column-stochastic context with potential applications in quantum information and statistical experiment theory.

Abstract

In this paper, we study majorization for probability distributions and column stochastic matrices. We show that majorizations in general can be reduced to the aforementioned sets. We characterize linear operators that preserve majorization for probability distributions, and show their equivalence to operators preserving vector majorization. Our main result provides a complete characterization of linear preservers of strong majorization for column stochastic matrices, revealing a richer structure of preservers than in the standard setting. As a prerequisite to this characterization, we solve the problem of characterizing linear preservers of majorization for zero-sum vectors, which yields a new structural insight into the classical results of Ando and of Li and Poon.

Majorizations for probability distributions, column stochastic matrices and their linear preservers

TL;DR

The paper addresses linear preservers of majorization for probability distributions and column stochastic matrices, establishing reductions to column-stochastic settings and solving a prerequisite zero-sum vector problem. It proves that preserving majorization on probability distributions is equivalent to preserving vector majorization, via reductions to -vectors, and then delivers a complete characterization of linear preservers of strong majorization for column stochastic matrices. A key technical contribution is the description of preservers on the zero-sum space, showing they must have the form , which links to Ando-type preservers and clarifies the structure of linear preservers in this setting. The main result unifies Li–Poon and Ando-type operators within a single structural form, providing a richer preservers theory for matrix majorization in the column-stochastic context with potential applications in quantum information and statistical experiment theory.

Abstract

In this paper, we study majorization for probability distributions and column stochastic matrices. We show that majorizations in general can be reduced to the aforementioned sets. We characterize linear operators that preserve majorization for probability distributions, and show their equivalence to operators preserving vector majorization. Our main result provides a complete characterization of linear preservers of strong majorization for column stochastic matrices, revealing a richer structure of preservers than in the standard setting. As a prerequisite to this characterization, we solve the problem of characterizing linear preservers of majorization for zero-sum vectors, which yields a new structural insight into the classical results of Ando and of Li and Poon.

Paper Structure

This paper contains 16 sections, 66 theorems, 49 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Reduction to column stochastic matrices
  4. Strong majorization
  5. Weak majorization
  6. Directional majorization
  7. Vector majorization
  8. $(0, 1)$-matrices
  9. Linear operators preserving majorization for probability distributions
  10. Linear operators preserving majorization for the zero-sum vectors
  11. Matrices satisfying $(\alpha)$
  12. Characterization of linear operators preserving majorization for the zero-sum vectors
  13. Linear operators preserving strong majorization for column stochastic matrices
  14. Every $\Phi^j_k$ preserves majorization on $\mathbbold{0}^n$
  15. Permutation matrices of $\Phi^j_k$ coincide
  16. ...and 1 more sections

Key Result

Theorem 1.1

MaOlAr11 The elements of the set of permutation matrices $P(n)$ are the extreme points of $\Omega_n$. Moreover, $\Omega_n$ is the convex hull of matrices in $P(n)$.

Theorems & Definitions (131)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 2.1
  • Lemma 2.2
  • ...and 121 more