IJPAM: Volume 32, No. 4 (2006)

LINEAR PRESERVER OF MATRIX MAJORIZATION

A.M. Hasani$^1$, M. Radjabalipour$^2$
$^{1,2}$Department of Mathematics
University of Kerman
Kerman, IRAN
$^1$e-mail: mohamad.h@graduate.uk.ac.ir
$^2$e-mail: radjab45@mail.uk.ac.ir


Abstract.An $n\times m$ matrix $A$ is said to be matrix majorized from the left by an $n\times m$ matrix $B$, and write $A\prec_{\ell} B$, if there exists an $n\times n$ row stochastic matrix $R$ such that $A=RB$. Let $M_{nm}$ denote the linear space of all real $n\times m$ matrices. An operator $T:M_{nm} \longrightarrow M_{nm}$ is said to be a preserver of $\prec_{\ell}$ if $TX \prec_{\ell} TY$ whenever $X\prec_{\ell} Y$ and $X,Y\in M_{nm}$. It is shown that a linear operator $T:M_{nm} \longrightarrow M_{nm}$ preserves $\prec_{\ell}$ if and only if there exist an $n\times n$ permutation matrix $P\neq I$, an $m\times m$ real matrix $L$, and real numbers $a$ and $b$ with $ab\leq 0$, such that $TX= (aI+bP)XL$ for all $X\in M_{nm}$ and, if $n\neq 2$, $ab=0$. Moreover, if $T$ satisfies the extra condition $X\prec_{\ell} Y$ whenever $TX \prec_{\ell} TY$, then $ab=0$ for all $n$ and $aI+bP$ and $L$ are invertible.

Received: September 25, 2006

AMS Subject Classification: 15A04, 15A21, 15A30

Key Words and Phrases: row stochastic matrix, matrix majorization, linear preserver

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 32
Issue: 4