Document Type: Research Paper

**Authors**

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Zip Code: 7718897111, Rafsanjan, Iran.

**Abstract**

In this paper, we study some kinds of majorizations on $\textbf{M}_{n}$ and their linear or strong linear preservers. Also, we find the structure of linear or strong linear preservers which are multiplicative, i.e. linear or strong linear preservers like $\Phi $ with the property $\Phi (AB)=\Phi (A)\Phi (B)$ for every $A,B\in \textbf{M}_{n}$.

**Keywords**

[1] T. Ando, Majorization, *Doubly stochastic matrices, and comparision of eigenvalues*, Linear Algebra and its Applications, 118 (1989) 163-248.

[2] A. Armandnejad, F. Akbarzadeh, and Z. Mohammadi, *Row and column-majorization on M_{n;m}*, Linear Algebra and its Applications, 437 (2012) 1025-1032.

[3] A. Armandnejad and H. Heydari, *Linear Preserving gd-Majorization Functions from M_{n;m} to M_{n;k}*, Bull. Iranian Math. Soc., 37(1) (2011) 215-224.

[4] A. Armandnejad and A. Ilkhanizadeh Manesh, *gut-Majorization and its Linear Preservers*, Electronic Journal of Linear Algebra, 23 (2012) 646-654.

[5] A. Armandnejad, Z. Mohammadi, and F. Akbarzadeh, *Linear preservers of G-row and G-column majorization on M_{n;m}*, Bull. Iranian Math. Soc., 39(5) (2013) 865-880.

[6] A.M. Hasani and M. Radjabalipour, *The structure of linear operators strongly preserving majorizations of matrices*, Electronic Journal of Linear Algebra, 15 (2006) 260-268.

[7] C.K. Li and E. Poon, *Linear operators preserving directional majorization*, Linear Algebra and its Applications, 325 (2001) 15-21.

[8] P. Semrl, *Maps on matrix spaces*, Linear Algebra and its Applications, 413(2-3) (2006) 364-393.