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Continuous Adjacency Preserving Maps on Real Matrices

Published:2005-06-01
Printed: Jun 2005
• Leiba Rodman
• Peter Šemrl
• Ahmed R. Sourour
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Abstract

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.
 Keywords: adjacency of matrices, continuous preservers, affine transformations
 MSC Classifications: 15A03 - Vector spaces, linear dependence, rank 15A04. - unknown classification 15A04.

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