Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Linear Preservers of Ultraweak, Row, and Weakly Hadamard Majorization on \(M_{mn}\)

  • 1 Accesses


Let \(X,Y\in M_{mn}\). We say that X is ultraweak Hadamard majorized by Y, denoted by \(X\prec _H^{uw} Y\), if there exists a matrix \(D=[d_{ij}]\in M_{mn}\), where \(0\le d_{ij}\le 1\), such that \(X=D\circ Y\). Also, we say that X is row Hadamard majorized (resp. weakly Hadamard majorized) by Y, denoted by \(X \prec ^{r}_{H} Y\) (resp. \(X \prec ^{w}_{H} Y\)), if there exists a row stochastic matrix R (resp. doubly substochastic matrix D), such that \(X=R\circ Y\)(resp. \(X=D\circ Y\)). In this paper, some properties of \( \prec _H^{uw}, \prec ^{r}_{H}\) and \(\prec ^{w}_{H}\) on \(M_{mn}\) are first obtained, and then, the (strong) linear preservers of \( \prec _H^{uw}, \prec ^{r}_{H}\) and \(\prec ^{w}_{H}\) on \(M_{mn}\) are characterized.

This is a preview of subscription content, log in to check access.


  1. 1.

    Ando, T.: Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl. 118, 163–248 (1989)

  2. 2.

    Beasley, L.B., Lee, S.G., Lee, Y.H.: A characterization of strong preservers of matrix majorization. Linear Algebra Appl. 367, 341–346 (2003)

  3. 3.

    Bhatia, R.: Matrix Analysis. Springer-Verlag, New York (1997)

  4. 4.

    Dahl, G.: Matrix majorization. Linear Algebra Appl. 288, 53–73 (1999)

  5. 5.

    Hasani, A.M., Radjabalipour, M.: On linear preservers of (right) matrix majorization. Linear Algebra Appl. 423, 255–261 (2007)

  6. 6.

    Hasani, A.M., Radjabalipour, M.: The structure of linear operators strongly preserving majorizations of matrices. Electron. J. Linear Algebra 15, 260–268 (2006)

  7. 7.

    Hasani, A.M., Vali, M.A.: Linear maps which preserve or strongly preserve weak majorization, J. Inequal. Appl., 2007, Article ID 82910 (2007)

  8. 8.

    Li, C.K., Poon, E.: Linear operators preserving directional majorization. Linear Algebra Appl. 325, 15–21 (2001)

  9. 9.

    Li, C.K., Tam, B.S., Tsing, N.K.: Linear maps preserving permutation and stochastic matrices. Linear Algebra Appl. 341, 5–22 (2002)

  10. 10.

    Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorizations and its Applications, 2nd edn. Springer, New York (2001)

  11. 11.

    Martínez Pería, F.D., Massey, P.G., Silvestre, L.E.: Weak matrix-majorization. Linear Algebra Appl. 403, 343–368 (2005)

  12. 12.

    Motlaghian, S.M., Armandnejad, A., Hall, F.J.: Linear preservers of Hadamard majorization. Electron. J. Linear Algebra 31, 593–609 (2016)

Download references


Thanks to the referee and the editor for their comments.

Author information

Correspondence to Ahmad Mohammadhasani.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Abbas Salemi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mohammadhasani, A. Linear Preservers of Ultraweak, Row, and Weakly Hadamard Majorization on \(M_{mn}\). Bull. Iran. Math. Soc. (2020).

Download citation


  • Ultraweak Hadamard majorization
  • Row Hadamard majorization
  • Weakly Hadamard majorization
  • linear preserver

Mathematics Subject Classification

  • 15A04
  • 15A21
  • 15A51