Vali-e-Asr university of RafsanjanWavelet and Linear Algebra2383-19368220220301Decomposability of Weak Majorizationتجزیه پذیری مهتری ضعیف111825114710.22072/wala.2021.525980.1321ENFatemehKhalooeiShahid Bahonar University of KermanAsmaIlkhanizadeh ManeshAssistant Professor, Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box: 7713936417, Rafsanjan, IranJournal Article20210227Let $x, y\in \mathbb{R}^n.$ 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 $x\prec_w y$.Let x,y ∈Rn. Weusethenotation x ≺w y when x isweakly majorized by y. We say that x ≺w y is decomposable at k (1 ≤ k < n) if x ≺w y has a coincidence at k and yk ̸= yk+1. Corresponding to this majorization we have a doubly substochastic matrix P. The paper presents x ≺w y is decomposable at some k (1 ≤ k < n) if and only if P is of the form D⊕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 ≺w y.https://wala.vru.ac.ir/article_251147_c619678d79cc8395799bf41552b73be2.pdf