IJPAM: Volume 60, No. 3 (2010)

PRESERVATION OF THE RANK OF
MATRICES FORCES THE LINEARITY

Józef Kalinowski
Department of Mathematics
Silesian University
14, Ul. Bankowa, Katowice, 40-007, POLAND
e-mail: kalinows@ux2.math.us.edu.pl


Abstract.Operators preserving the rank of real matrices were studied in Basley [#!2!#] under assumption that the operator is linear. In the present paper the linearity of the operator is not assumed: we assume only that the operator is of the form $F=[f_{i,j}]$, where $f_{i,j}:\R\longrightarrow \R$ are functions for $i=1,2,\dots,m;$ $j=1,2,\dots,n$. If the min{$m,n\}\geq 3$, then in the matrix space $M_{m,n}$ the operator preserving the rank of matrices must be linear as in Basley [#!2!#]. If the min$\{m,n\}\leq 2$, then the operator may be nonlinear. In both cases the forms of the operator are presented.

Received: February 11, 2010

AMS Subject Classification: 15A03

Key Words and Phrases: rank of matrices, preservers

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 60
Issue: 3