Continuous Adjacency Preserving Maps on Real Matrices

http://dx.doi.org/10.4153/CMB-2005-025-6
Canad. Math. Bull. 48(2005), 267-274

  Published:2005-06-01
 Printed: Jun 2005

It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form $ A \mapsto PAQ+R$, possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix $R$. The result extends previously known theorems where the map was assumed to be also injective.