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Linear operators preserving majorization of matrix tuples
- Source :
- Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy. 7:217-229
- Publication Year :
- 2020
- Publisher :
- Saint Petersburg State University, 2020.
-
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 Ax is majorized by Bx for any vector x where the usual vector majorization is used. We introduce the notion of majorization of matrix tuples which is defined as a natural generalization of matrix majorizations: for a chosen type of majorization we say that one tuple of matrices is majorized by another tuple of the same size if every matrix of the “smaller” tuple is majorized by a matrix in the same position in the “bigger” tuple. We say that a linear operator preserves 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 of tuples of matrices and linear operators that map tuples that are ordered with respect to strong majorization to tuples that are ordered with respect to directional majorization. We have shown that every such operator preserves respective majorization of each component. For all types of majorization we provide counterexamples that demonstrate that the inverse statement does not hold, that is if majorization of each component is preserved, majorization of tuples may not.
- Subjects :
- Doubly stochastic matrix
General Mathematics
010102 general mathematics
Stochastic matrix
General Physics and Astronomy
01 natural sciences
Square matrix
010305 fluids & plasmas
Combinatorics
Linear map
Matrix (mathematics)
0103 physical sciences
Ordered pair
0101 mathematics
Tuple
Majorization
Mathematics
Subjects
Details
- ISSN :
- 25875884 and 10253106
- Volume :
- 7
- Database :
- OpenAIRE
- Journal :
- Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy
- Accession number :
- edsair.doi...........5f129026a33bc9ddb9069084d383a93c