Radjabalipuor,The structure of linear operators strongly preserving majorizations of matrices

Abstract. A matrix majorization relation A ≺r B (resp., A ≺ℓ B) on the collection Mn of all n × n real matrices is a relation A = BR (resp., A = RB) forsomen × 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 ≺rmul B (resp., A ≺ℓmul B) isarelationA = BD (resp., A = DB) forsomen × 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 when R(T (X),T(Y)) if and only if R(X, Y). The results are the sharpening of well-known representations TX = CXD or TX = CXtD for linear operators preserving invertible matrices

Linear preservers of left matrix majorization

Abstract. For X, Y ∈ Mnm(R) (=Mnm), we say that Y is left (resp. right) matrix majorized by X and write Y ≺ℓ X (resp. Y ≺r X) ifY = RX (resp. Y = XR) for some row stochastic matrix R. A linear operator T: Mnm → Mnm is said to be a linear preserver of a given relation ≺ on Mnm if Y ≺ X implies that TY ≺ TX. The linear preservers of ≺ℓ or ≺r are fully characterized by A.M. Hasani and M. Radjabalipour. Here, we launch an attempt to extend their results to the case where the domain and the codomain of T are not necessarily identical. We begin by characterizing linear preservers T: Mp1 → Mn1 of ≺ℓ