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 $n\geq 2$ and let $\Phi:S_{n}\ra S_{m}$ be a map
preserving adjacency, i.e. if $A,B\in S_{n}$ and $\hbox{rank}~(A-B)=1$,
then $\hbox{rank}~(\Phi(A)-\Phi(B))=1$. If $ \Phi(0)=0$,
we prove that either:
(i) $\Phi$ maps $S_n$ into $\rr B$, where $B$ is a rank one matrix, or (ii) there exist $c \in \left\{ -1,1 \right\}$ and $R \in M_m$ invertible ($ m \geq n $) such that for $A\in S_n$,
\[
\Phi(A)=cR
\left[
\begin{array}{cc}
A&0 \\
0&0
\end{array}
\right]R^T.
\]
(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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