Majorization: Here, There and Everywhere
نویسنده
چکیده
The appearance of Marshall and Olkin’s 1979 book on inequalities with special emphasis on majorization generated a surge of interest in potential applications of majorization and Schur convexity in a broad spectrum of fields. After 25 years this continues to be the case. The present article presents a sampling of the diverse areas in which majorization has been found to be useful in the past 25 years.
منابع مشابه
SGLT-MAJORIZATION ON Mn,m AND ITS LINEAR PRESERVERS
A matrix R is said to be g-row substochastic if Re ≤ e. For X, Y ∈ Mn,m, it is said that X is sglt-majorized by Y , X ≺sglt Y , if there exists an n-by-n lower triangular g-row substochastic matrix R such that X = RY . This paper characterizes all (strong) linear preservers and strong linear preservers of ≺sglt on Rn and Mn,m, respectively.
متن کاملMajorization, Polyhedra and Statistical Testing Problems. Majorization, Polyhedra and Statistical Testing Problems
There are important connections between majorization and convex polyhedra. Both weak majorization and majorization are preorders related to certain simple convex cones. We investigate the facial structure of a polyhedral cone C associated with a layered directed graph. A generalization of weak majorization based on C is introduced. It de nes a preorder of matrices. An application in statistical...
متن کاملLinear preservers of Miranda-Thompson majorization on MM;N
Miranda-Thompson majorization is a group-induced cone ordering on $mathbb{R}^{n}$ induced by the group of generalized permutation with determinants equal to 1. In this paper, we generalize Miranda-Thompson majorization on the matrices. For $X$, $Yin M_{m,n}$, $X$ is said to be Miranda-Thompson majorized by $Y$ (denoted by $Xprec_{mt}Y$) if there exists some $Din rm{Conv(G)}$ s...
متن کاملBlock Diagonal Majorization on $C_{0}$
Let $mathbf{c}_0$ be the real vector space of all real sequences which converge to zero. For every $x,yin mathbf{c}_0$, it is said that $y$ is block diagonal majorized by $x$ (written $yprec_b x$) if there exists a block diagonal row stochastic matrix $R$ such that $y=Rx$. In this paper we find the possible structure of linear functions $T:mathbf{c}_0rightarrow mathbf{c}_0$ preserving $prec_b$.
متن کاملWeak matrix majorization
Given X, Y ∈ Rn×m we introduce the following notion of matrix majorization, called weak matrix majorization, X w Y if there exists a row-stochastic matrix A ∈ Rn×n such that AX = Y, and consider the relations between this concept, strong majorization ( s ) and directional majorization ( ). It is verified that s⇒ ⇒ w , but none of the reciprocal implications is true. Nevertheless, we study the i...
متن کامل