Continuous Adjacency Preserving Maps on Real Matrices
Printed: Jun 2005
Ahmed R. Sourour
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.
adjacency of matrices, continuous preservers, affine transformations
15A03 - Vector spaces, linear dependence, rank
15A04. - unknown classification 15A04.