Weak matrix majorization

Weak 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...

On Nonlinear Preservers of Weak Matrix Majorization

on nonlinear preservers of weak matrix majorization

Matrix majorization Geir Dahl

We study the conceptmatrix majorization: for two real matrices A and B having m rows we say that A majorizes B if there is a row-stochastic matrix X with AX = B. A special case is classical notion of vector majorization. Several properties and characterizations of matrix majorization are given. Moreover, interpretations of the concept in mathematical statistics are discussed and some combinator...

Weak k-majorization and polyhedra

For integers k and n with k ≤ n a vector x ∈ R is said to be weakly k-majorized by a vector q ∈ R if the sum of the r largest components of x does not exceed the sum of the r largest components of q, for r = 1, . . . , k. For a given q the set of vectors weakly k-majorized by q defines a polyhedron P (q; k). We determine the vertices of both P (q; k) and its integer hull Q(q; k). Furthermore a ...