Generalized row substochastic matrices and majorization

Document Type : Original Paper


Department of Mathematics, Sirjan University Of Technology, Sirjan, Iran


‎The square and real matricx $A$ is called a generalized row substochastic matrix‎, ‎if the sum of the absolute values of the entries in each row is less than or equal to one‎.

‎Let $x,y\in \mathbb{R}^n$‎. ‎We say that $x$ is right $B$-majorized (resp‎. ‎left $B$-majorized) by $y$‎, ‎denoted by $x \prec _{rB} y$ ($x \prec _{lB} y$)‎, ‎if there exists a substochastic matrix $D$‎, ‎such that $x=yD$ (resp‎. ‎$x=Dy$)‎. ‎In this article‎, ‎we have found all the vectors such as $x$ that $x$ is right $B$-majorized (resp‎. ‎left $B$-majorized) by $y$‎, ‎for all row vector $y$ (resp‎. ‎column vector)‎. ‎Also‎, ‎we show $x \sim _{lB} y$ if and only if $\Vert x\Vert_\infty =\Vert y\Vert_\infty$ and prove $x \sim _{rB} y$ if and only if $\Vert x\Vert_1 =\Vert y\Vert_1$‎.

‎We have also created conditions in which the left $B$-majorization is equivalent to the left majorization‎, ‎and created conditions in which the right $B$-majorization is equivalent to the right majorization‎.


Main Subjects

[1] ع. آرمندنژاد، مروری بر مهتری های عادی و تعمیم یافته و بررسی ساختار نگه دارنده های خطی آن ها، فرهنگ و اندیشه ریاضی، 45، (1389) 31-40.
[2] ا. محمدحسنی، ی. سیاری و م. سبزواری، نگه دارنده های خطی راست-چپ ماتریسی، موجک ها و جبر خطی،  8 (3) (1401) 37-59.
[3] A. Armandnejad and Z. Gashool, Strong linear preservers of g-tridiagonal majorization on Rn. Electronic Journal of Linear Algebra, 123 (2012) 115-121.
[4] A. Armandnejad, S. Mohtashami, and M. Jamshidi, On linear preservers of g-tridiagonal majorization on Rn. Linear Algebra and its Applications, 459 (2014) 145-153.
[5] R. Bahatia, Matrix Analysis. Springer-Verlag, New York, 1997.
[6] L. B. Beasley, S-G. Lee and Y-H Lee, A characterization of strong preservers of matrix majorization. Linear Algebra and its Applications, 367 (2003) 341-346.
[7] R. A. Brualdi and G. Dahl, An extension of the polytope of doubly stochastic matrices, Linear and Multilinear Algebra, 6 (3) (2013) 393-408.
[8] H. Chiang and C. K. Li, Generalized doubly stochastic matrices and linear preservers. Linear and Multilinear Algebra, 53 (2005) 1-11.
[9] G. Dahl, Matrix majorization, Linear Algebra Appl., 288 (1999) 53-73.
[10] Francisco D. Martínez Pería, Pedro G. Massey, Luis E. Silvestre, Weak matrix majorization, Linear Algebra and its Applications 403 (2005) 343–368.
[11] M.H. Hadian, A. Armandnejad, B-majorization and its linear preservers, Linear Algebra and its Application, 478 (2015) 218-227.
[12] A. M. Hasani, A. Ilkhanizadeh Manesh, Linear preservers of two-sided right matrix majorization on Rn, Adv. Oper. Theory, 3 (2018) 1-8
[13] A. M. Hasani and M. Radjabalipour, The structure of linear operators strongly preservingmajorizations of matrices.Electronic Journal of Linear Algebra, 15 (2006) 260-268.
[14] M. Marcus, All linear operators leaving the unitary group invariant, Duke Math. J. 26 (1959) 155-163.
[15] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of majorization and its applications. Springer, New York, 2011.
Volume 12, Issue 4 - Serial Number 28
January 2022
Pages 523-534
  • Receive Date: 11 March 2021
  • Revise Date: 26 December 2022
  • Accept Date: 28 December 2022
  • First Publish Date: 28 December 2022