Linear Preservers of Chain Majorization

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 ¯nitely 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 ??.