Decomposability of Weak Majorization

Document Type : Research Paper


Shahid Bahonar University of Kerman



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 $x\prec_w y$.


[1] T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl., 118, (1989), 
[2] A. Armandnejad and A. Ilkhanizade manesh, Gut-majorization on $mathbb{M}_{n,m}$ and its linear preservers, 
     Electron. J. Linear Algebra, 23, (2012), 646-654.
[3] R. Bhatia, Matrix Analysis, Springer, New York, 1997.
[4] R.A. Brualdi, The doubly stochastic matrices of a vector majorization, Linear Algebra Appl., 61, (1984), 141-154.
[5] G. Dahl and F. Margot, Weak k-majorization and polyhedra, Math. Program., 81, (1998), 37-53.
[6] A. Mohammadhasani  and A. Ilkhanizade manesh, Linear preservers of right sgut-majorization on $mathbb{R}^n$, 
     Wavel. Linear Algebra, 3, (2017), 116-133.
[7] F. Khalooei, Linear preservers of two-sided matrix majorization, Wavelets and Linear Algebra, 1, (2014), 43-50.
[8] F. Khalooei and A. Salemi, The structure of linear preservers of left matrix majorization on $mathbb{R}^p$, Electron. J. 
      Linear Algebra, 18, (2009), 88-97.
[9] R.B. Levow, A problem of mirsky concerning nonsingular doubly stochastic matrices, Linear Algebra Appl., 5, (1972), 
[10] A.W. Marshall, I. Olkin and B.C. Arnold,  Inequalities: Theory of Majorization and Its Applications, Springer, New York,