# Linear Preservers of Doubly stochastic matrices and permutation matrices from $M_m$ to $M_n$

Document Type : Research Paper

Authors

1 Instietute of Advanced Studies, Payame Noor University, P.O. Box 19395- 3697, Tehran, Iran.

2 Department of Mathematic, Payame Noor University, P.O. Box 19395- 3697, Tehran, Iran.

3 Department of Pure Mathematics, Faculty of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, 88186-34141, Iran.

10.22072/wala.2021.131790.1298

Abstract

Chi-Kwang Li, Bit-Shun Tam and Nam-Kiu Tsing have obtained necessary and sufficient condition  for a linear operator on linear space of generalized doubly stochastic matrices to be strong preserver of doubly stochastic matrices and permutation matrices.
We show if a linear operator $T:M_m\rightarrow M_n$ is a (strong) preserver of doubly stochastic matrices, then $T$ is a (strong) preserver of the linear manifold of r-generalized doubly stochastic matrices and the linear space of generalized doubly stochastic matrices. Also we give necessary and sufficient condition for a linear operator $T:M_m\rightarrow M_n$ to be (strong) preserver of doubly stochastic matrices and permutation matrices.

Keywords

#### References

[1] A. Douik and B. Hassibi, Manifold optimization over the set of doubly stochastic matrices, IEEE Trans. Signal Process., 67(22) (2019), 5761-5774.
[2] D. Hug and W. Weil, A Course on Convex Geometry, University of Karlsruhe 2011.
[3] C.K. Li, B.S. Tam and N.K. Tsing, Linear maps preserving permutation and stochastic matrices, Linear Algebra Appl., 341 (2002), 5-22.
[4] A.W. Marshall, I. Olkin and B.C. Arnold, Inequalities, Theory of Majorization and Its Applications, 2nd ed., Springer, NewYork, 2011.
[5] B. Mourad, On lie theoretic approach to general doubly stochastic matrices and applications, Linear Multilinear Algebra, 52(2) (2013), 99-113.
[6] M. Murray, Some Notes on Differential Geometry, University of Adelaide July 20, 2009.