doubly stochastic matrix

Chi-Kwang Li, Bit-Shun Tam and Nam-Kiu Tsing have obtained necessary and sufficient condition for a linear operator on linear space of generalized doubly stochastic matrices to be strong preserver of doubly stochastic matrices and permutation matrices. We show if a linear operator T : Mm → Mn is a (strong) preserver of doubly stochastic matrices, then T is a (strong) preserver of the linear manifold of r-generalized doubly stochastic matrices and the linear space of generalized doubly stochastic matrices. Also we give necessary and sufficient condition for a linear operator T : Mm → Mn to be (strong) preserver of doubly stochastic matrices and permutation matrices.

In this paper we study the concept of Latin-majorizati-\\on. Geometrically this concept is different from other kinds of majorization in some aspects. Since the set of all xs Latin-majorized by a fixed y is not convex, but, consists of union of finitely many convex sets. Next, we hint to linear preservers of Latin-majorization on Rn and Mn,m.

In this paper, we study some kinds of majorizations on Mn and their linear or strong linear preservers. Also, we find the structure of linear or strong linear preservers which are multiplicative, i.e. linear or strong linear preservers like Φ with the property Φ(AB)=Φ(A)Φ(B) for every A,B∈Mn.

For (n×m matrices) X, Y ∈ 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 T X ≺≺ T Y . Also, it is said to be strong linear preserver if X ≺≺ Y is equivalent to T X ≺≺ T Y . In this paper we characterize linear and strong linear preservers of ≺≺.