Linear mappings of local preserving-majorization on matrix algebras

TL;DR

This paper characterizes when linear maps preserve majorization relations on matrices, showing that strictly decreasing vectors are special points where all isotone maps behave consistently.

Contribution

It proves that vectors with strictly decreasing entries are strictly all-isotone points for linear maps preserving majorization.

Findings

Vectors with strictly decreasing entries are strictly all-isotone points.

Characterization of linear maps that preserve majorization.

Insight into the structure of isotone mappings on matrix algebras.

Abstract

Let $\M_{n\times n}$ be the algebra of all $n\times n$ matrices. For $x,y\in {R}^{n}$ it is said that $x$ is majorized by $y$ if there is a double stochastic matrix $A\in {M}_{n\times n}$ such that $x=Ay$ (denoted by $x\prec y$). Suppose that $\Phi$ is a linear mapping from ${R}^{n}$ into ${R}^{n}$, which is said to be strictly isotone if $\Phi(x)\prec \Phi(y)$ whenever $x\prec y$. We say that an element $\alpha\in {R}^{n}$ is a strictly all-isotone point if every strictly isotone $\varphi$ at $\alpha$ (i.e. $\Phi(\alpha)\prec\Phi(y)$ whenever $x\in {R}^{n}$ with $\alpha\prec x$, and $\Phi(x)\prec\Phi(\alpha)$ whenever $x\in {R}^{n}$ with $x\prec \alpha$) is a strictly isotone. In this paper we show that every $\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{n})\in {R}^{n}$ with $\alpha_{1}>\alpha_{2}>...>\alpha_{n}$ is a strictly all-isotone point.