Let F and Fm be the usual spaces of n-dimensional column and m-dimensional row vectors on F, respectively, where F is the field of real or complex numbers. In this paper, the relations gsmajorization, lgw-majorization, and rgw-majorization are considered on F and Fm. Then linear maps T : F → F preserving lgw-majorization or gs-majorization and linear maps S : Fn → Fm, preserving rgw-majorizatio...

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...

For many least-squares decomposition models efficient algorithms are well known. A more difficult problem arises in decomposition models where each residual is weighted by a nonnegative value. A special case is principal components analysis with missing data. Kiers (1997) discusses an algorithm for minimizing weighted decomposition models by iterative majorization. In this paper, we propose a m...

In this paper we show how the Shannon entropy is connected to the theory of majorization. They are both linked to the measure of disorder in a system. However, the theory of majorization usually gives stronger criteria than the entropic inequalities. We give some generalized results for majorization inequality using Csiszár f-divergence. This divergence, applied to some special convex functions...

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...

We generalize the classical notion of majorization in Rn to a majorization order for functions defined on a partially ordered set P . In this generalization we use inequalities for partial sums associated with ideals in P . Basic properties are established, including connections to classical majorization. Moreover, we investigate transfers (given by doubly stochastic matrices), complexity issue...

We introduce a partial order, variance majorization, on R, which is analogous to the majorization order. A new class of monotonicity inequalities, based on variance majorization and analogous to Schur convexity, is developed.