title: Linear operators preserving majorization of matrix tuples
metadata.name: dc.contributor.author
metadata.value: Guterman, Alexander E.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Shteyner, Pavel M.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2020-05-28T14:01:30Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2020-05-28T14:01:30Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2020-06
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Guterman A. E., Shteyner P.M. Linear operators preserving majorization of matrix tuples. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2020, vol. 7 (65), issue 2, pp. 217–229.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/11701/spbu01.2020.204
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/17931
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: 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.
metadata.lang: en_GB
metadata.name: dc.description.sponsorship
metadata.value: The work is supported by Russian Science Foundation (grant N16-11-10075).
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy;Volume 7 (65); Issue 2
metadata.lang: -
metadata.name: dc.subject
metadata.value: matrix majorization
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: vector majorization
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: linear preservers
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Linear operators preserving majorization of matrix tuples
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 217-229.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/17931/1/217-229.pdf
resources.format: Adobe PDF
resources.size: 349,79 kB