Linear operators preserving majorization of matrix tuples

Abstract

In this paper, we consider weak, directional and strong matrix majorizations. Namely, for square matrices A and B of the same size we say that A is weakly majorized by B, if there is a row stochastic matrix X such that A = XB. Further, A is strongly majorized by B, if there is a doubly stochastic matrix X such that A = XB. Finally, A is directionally majorized by B, if for any vector x the vector Ax is majorized by the vector Bx under the usual vector majorization. We introduce the notion of majorization for matrix tuples, which is defined as a natural generalization of matrix majorizations: for a chosen type of majorization we say that one matrix tuple is majorized by another matrix tuple of the same size if every matrix of the “smaller” tuple is majorized by the matrix in the same position in the “bigger” tuple. We say that linear operator preserves a majorization if it maps ordered pairs to ordered pairs and the image of the smaller element does not exceed the image of the bigger one. This paper contains a full characterization of linear operators that preserve weak, strong or directional majorization for matrix tuples, as well as linear operators that map matrix tuples ordered with respect to the strong majorization to matrix tuples ordered with respect to the directional majorization. We have shown that every such operator preserves respective majorization for each component. For each of these three types of majorizations we provide counterexamples demonstrating that the inverse statement does not hold, that is, if majorization of each component is preserved, majorization of matrix tuples may not be preserved.