Original scientific paper

Adjacency preserving mappings on real symmetric matrices

Peter Legiša ; Department of Mathematics, FMF, University of Ljubljana, Ljubljana, Slovenia

Abstract

Let $S_n$ denote the space of all $n\times n$ real symmetric matrices.
Let $n\ge 2$ and let $\mathrm{\Phi }:S_n\text{ra}{S}_m$ be a map
preserving adjacency, i.e. if $A,B\in S_n$ and $\text{rank}(A-B)=1$,
then $\text{rank}(\mathrm{\Phi }(A)-\mathrm{\Phi }(B))=1$. If $\mathrm{\Phi }(0)=0$,
we prove that either:
(i) $\mathrm{\Phi }$ maps $S_n$ into $\text{rr}B$, where $B$ is a rank one matrix, or (ii) there exist $c\in \{-1,1\}$ and $R\in M_m$ invertible ($m\ge n$) such that for $A\in S_n$,
$$\mathrm{\Phi }(A)=cR\left[\begin{array}{cc}A& 0\\ 0& 0\end{array}\right]R^T.$$
(If $m=n$, the zeros on the right-hand side are absent.)

Keywords

real symmetric matrix; adjacency preserving map; rank; geometry of matrices

Hrčak ID:

74883

URI

https://hrcak.srce.hr/74883

Publication date:

21.12.2011.

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