ELA, Volume 15, pp. 260-268, September 2006, abstract.

The Structure of Linear Operators Strongly Preserving
Majorizations of Matrices

Ahmad M. Hasani and Mehdi Radjabalipour

A matrix majorization relation A <_r B (resp., A <_l B) on the 
collection M_n of all nxn real matrices is a relation A=BR 
(resp., A=RB) for some nxn row stochastic matrix R (depending 
on A and B). These right and left matrix majorizations have 
been considered by some authors under the names "matrix 
majorization" and "weak matrix majorization," respectively.
Also, a multivariate majorization A <_rmul B (resp., A<_lmul B) 
is a relation A=BD (resp., A=DB) for some nxn doubly stochastic 
matrix D (depending on A and B). The linear operators T:M_n --> M_n 
which strongly preserve each of the above mentioned majorizations 
are characterized. Recall that an operator T:M_n --> M_n strongly 
preserves a relation IR on M_n when IR(T(X),T(Y)) if and only if
IR(X,Y). The results are the sharpening of well-known representations 
TX=CXD or TX=CX^tD for linear operators preserving invertible matrices.