A characterization of strong preservers of matrix majorization
The definition of matrix majorization was introduced by \textit{G. Dahl} [Linear Algebra Appl. 288, 53-73 (1999; Zbl 0930.15023)] given several characterizations. Let \(A\) and \(B\) be \(m\times n\) real matrices. The matrix \(A\) is said to be matrix majorized by \(B\), \(A\prec B\), if there is an \(n\times n\) row stochastic matrix \(X\) (a nonnegative real matrix in which each of its row sums are equal to one) such that \(A=BX\). This definition generalizes the concept of multivariate majorization and is the matrix version of vector majorization that plays an important role in statistics and economics. In this paper, the authors characterize linear operators \(T\) that strongly preserve matrix majorization (\(T(A)\prec T(B)\) iff \(A\prec B\)), extending the results of \textit{L. B. Beasley} and \textit{S.-G. Lee} [Linear Algebra Appl. 304, 141-159 (2000; Zbl 0954.15016)] over the characterization of linear operators that strongly preserve multivariate majorization. The main theorem is: If \(T:M_{n}(\mathbb{R})\rightarrow M_{n}(\mathbb{R})\) strongly preserves matrix majorization, then there exists a permutation matrix \(Q\) and an invertible matrix \(M\) such that \(T(A)=MAQ\) for every \(A\) belonging to the span of the set of all row stochastic matrices.
scientific article; zbMATH DE number 1917823