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 M_n 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 ≺_rmul B (resp., A ≺_ℓmul 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 : 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 R on M_n when R(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 = CX^tD for linear operators preserving invertible matrices.