Department of Mathmatics, Faculty of Mathmatics, Shahid Bahonar University of Kerman, Kerman, Islamic Republic of Iran
For vectors X, Y ∈ Rn, it is said that X is left matrix majorized by Y if for some row stochastic matrix R; X = RY. The relation X ∼` Y, is defined as follows: X ∼` Y if and only if X is left matrix majorized by Y and Y is left matrix majorized by X. A linear operator T : Rp → Rn is said to be a linear preserver of a given relation ≺ if X ≺ Y on Rp implies that T X ≺ TY on Rn. The linear preservers of ≺` from Rp to Rn are characterized before. In this parer we characterize the linear preservers of ∼` from Rp to Rn, p ≥ 3. In fact we show that the linear preservers of ∼` from Rp to Rn are the same as the linear preservers of ≺` from Rp to Rn, but for p = 2, they are not the same.
 T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl., 118 (1989), 163-248.  A. Armandnejad, F. Akbarzadeh and Z. Mohamadi,Row and column-majorization on Mn,m, Linear Algebra Appl., 437(2012), 1025-1032.  L. B. Beasley, S.-G. Lee and Y.-H. Lee, A characterization of strong preservers of matrix majorization, Linear Algebra Appl. 367 (2003), 341-346.  R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.  R. A. Brualdi and G. Dahl,Majorization classes of integral matrices, Linear Algebra Appl., 436(2012), 802-813.  A. M. Hasani and M. Radjabalipour,Linear preserver of matrix majorization, International Journal of Pure and Applied Mathematics, 32(4) (2006), 475-482.  A. M. Hasani and M. Radjabalipour,On linear preservers of (right) matrix majorization, Linear Algebra Appl., 423(2-3)(2007), 255-261.  F. Khalooei, M. Radjabalipour and P. Torabian, Linear preservers of left matrix majorization, Electron. J. Linear Algebra, 17(2008), 304-315.  F. Khalooei and A. Salemi,The structure of linear preservers of left matrix majorization on Rp, Electron. J. Linear Algebra, 18(2009), 88-97.  F. Khalooei and A. Salemi,Linear preservers of majorization, Iranian Journal of Mathematical Siences and Informatics, 6(2) (2011), 43-50.  C. K. Li and E. Poon,Linear operators preserving directional majorization, Linear Algebra Appl., 325 (2001), 141-146.  F. D. Martınez Perıa, P. G. Massey, and L. E. Silvestre, Weak Matrix-Majorization, Linear Algebra Appl., 403(2005), 343-368.