Document Type : Research Paper

**Author**

Department of Mathmatics, Faculty of Mathmatics, Shahid Bahonar University of Kerman, Kerman, Islamic Republic of Iran

**Abstract**

For vectors *X*, *Y *∈ R*n*, 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 *: R*p *→ R*n *is said to be a linear preserver of a given relation ≺ if *X *≺ *Y *on R*p *implies that *T X *≺ *TY *on R*n*. The linear preservers of ≺` from R*p *to R*n *are characterized before. In this parer we characterize the linear preservers of ∼` from R*p *to R*n*, *p *≥ 3. In fact we show that the linear preservers of ∼` from R*p *to R*n *are the same as the linear preservers of ≺` from R*p *to R*n*, but for *p *= 2, they are not the same.

**Keywords**

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