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Row Hadamard majorization on Mm,n

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Abstract

An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let Mm,n be the set of all m × n real matrices. For A, B ∈ Mm,n, we say that A is row Hadamard majorized by B (denoted by A ≺ RHB) if there exists an m × n row stochastic matrix R such that A = R ο B, where X ο Y is the Hadamard product (entrywise product) of matrices X, Y ∈ Mm,n. In this paper, we consider the concept of row Hadamard majorization as a relation on Mm,n and characterize the structure of all linear operators T: Mm,n → Mm,n preserving (or strongly preserving) row Hadamard majorization. Also, we find a theoretic graph connection with linear preservers (or strong linear preservers) of row Hadamard majorization, and we give some equivalent conditions for these linear operators on Mn.

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Acknowledgments

In the initial version of the paper the proof of Theorem 1.1 was very long. The present proof is suggested by the anonymous referee. The authors thank the referee for the elegant proof of Theorem 1.1 and some other comments.

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Authors and Affiliations

  1. Department of Mathematics, Vali-e-Asr University of Rafsanjan, Imam Khomeini Square, 7718897111, Rafsanjan, Iran

    Abbas Askarizadeh & Ali Armandnejad

Authors
  1. Abbas Askarizadeh
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  2. Ali Armandnejad
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Correspondence to Abbas Askarizadeh.

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Askarizadeh, A., Armandnejad, A. Row Hadamard majorization on Mm,n. Czech Math J 71, 743–754 (2021). https://doi.org/10.21136/CMJ.2020.0081-20

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  • DOI: https://doi.org/10.21136/CMJ.2020.0081-20

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