Generalized row substochastic matrices and majorization

Document Type : Original Paper

Authors

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

Abstract

‎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.

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Main Subjects

References

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History

• Receive Date: 11 March 2021
• Revise Date: 26 December 2022
• Accept Date: 28 December 2022