For vectors X, Y ϵ Rn, we say X is left matrix majorized by Y and write X<eY if for some row stochastic matrix R, X=RY. Also, we write X~eY, when X<eY<eX. A linear operator T: Rp®Rn is said to be a linear preserver of a given relation<if X<Y on Rp implies that TX<TY on Rn. In this note we study linear preservers of ~e from Rp to Rn. In particular, we characterize all linear preservers of ~e from R2 to Rn, and also, all linear preservers of ~e from Rp to Rp.

For (n´m matrices) X, Y 2 Mnm (R) (=Mnm), we say X is chain majorized by Y and write X << Y if X=RY where R is a product of finitely many T-transforms. A linear operator T: Mnm ® Mnm is said to be a linear preserver of the relation << on Mnm if X << Y implies that TX << TY. Also, it is said to be strong linear preserver if X << Y is equivalent to TX << TY. In this paper we characterize linear and strong linear preservers of <<.

Let Mm, nMm, n be the set of all mm-by-nn real matrices. A matrix RR in Mm, nMm, n with nonnegative entries is called strictly sub row stochastic if the sum of entries on every row of RR is less than 1. For A, B∈ Mm, nA, B∈ Mm, n, we say that AA is strictly sub row Hadamard majorized by BB (denoted by A≺ SHB)A≺ SHB) if there exists an mm-by-nn strictly sub row stochastic matrix RR such that A=R∘ BA=R∘ B where X∘ YX∘ Y is the Hadamard product (entrywise product) of matrices X, Y∈ Mm, nX, Y∈ Mm, n. In this paper, we introduce the concept of strictly sub row Hadamard Majorization as a relation on Mm, nMm, n. Also, we find the structure of all linear operators T: Mm, n→ Mm, nT: Mm, n→ Mm, n which are preservers (resp. strong preservers) of strictly sub row Hadamard Majorization.

A matrix A is said to be multivariate majorized by a matrix B, written A ≺ B, if there exists a doubly stochastic matrix D such that A = BD. In the present paper, we obtain a totally ordered subset of Mnm which contains a given matrix A. Also, we show that the totality of all extreme points of the collection of all matrices which are multivariately majorized by a matrix A is the set of all matrices obtained by permuting the columns of A.

Let c0 be the real vector space of all real sequences which converge to zero. For every x, yÎc0, it is said that y is block diagonal majorized by x (written y<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: c0®c0 preserving <b.