# Decomposability of Weak Majorization

Document Type : Research Paper

Author

Shahid Bahonar University of Kerman

10.22072/wala.2021.525980.1321

Abstract

Let $x, y\in \mathbb{R}^n.$ We use the notation $x\prec_w y$ when $x$ is weakly majorized by $y$. We say that $x\prec_w y$ is decomposable at $k$ $(1\leq k < n)$ if $x\prec_w y$ has a coincidence at $k$ and $y_{k}\neq y_{k+1}$. Corresponding to this majorization we have a doubly substochastic matrix $P$. The paper presents $x\prec_w y$ is decomposable at some $k$ $(1\leq k<n)$ if and only if $P$ is of the form $D\oplus Q$ where $D$ and $Q$ are doubly stochastic and doubly substochastic matrices, respectively. Also, we write some algorithms to obtain $x$ from $y$ when $x\prec_w y$.

Keywords

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