Geodesic
The structure of linear operators strongly preserving majorizations of matrices
The electronic journal of linear algebra, Tome 15 (2006), pp. 260-268.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: A matrix majorization relation A OEr B (resp., A OE` B) on the collection Mn of all n * n real matrices is a relation A = BR (resp., A = RB) for some n * n 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 OErmul B (resp., A OE`mul B) is a relation A = BD (resp., A = DB) for some n * n doubly stochastic matrix D (depending on A and B). The linear operators T : Mn ! Mn which strongly preserve each of the above mentioned majorizations are characterized. Recall that an operator T : Mn ! Mn strongly preserves a relation R on Mn whenR (T (X), T (Y )) if and only if $R(X, Y )$. The results are the sharpening of well-known representations T X = CXD or T X = CXtD for linear operators preserving invertible matrices.
Classification : 15A04, 15A21, 15A51
Keywords: row stochastic matrix, doubly stochastic matrix, matrix majorization, weak matrix majorization, $Left(right)$ multivariate majorization, linear preserver 
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     title = {The structure of linear operators strongly preserving majorizations of matrices},
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Hasani, Ahmad M.; Radjabalipour, Mehdi. The structure of linear operators strongly preserving majorizations of matrices. The electronic journal of linear algebra, Tome 15 (2006), pp. 260-268. http://geodesic.mathdoc.fr/item/ELA_2006__15__a7/