Abstract view
Adjacency Preserving Maps on Hermitian Matrices


Published:20081001
Printed: Oct 2008
Wenling Huang
Peter \v Semrl
Abstract
Hua's fundamental theorem of the geometry of hermitian matrices
characterizes bijective maps on the space of all $n\times n$
hermitian matrices preserving adjacency in both directions.
The problem of possible improvements
has been open for a while. There are three natural problems here.
Do we need the bijectivity assumption? Can we replace the
assumption of preserving adjacency in both directions by the
weaker assumption of preserving adjacency in one direction only?
Can we obtain such a characterization for maps acting between the
spaces of hermitian matrices of different sizes? We answer all
three questions for the complex hermitian matrices, thus obtaining
the optimal structural result for adjacency preserving maps on
hermitian matrices over the complex field.
MSC Classifications: 
15A03, 15A04, 15A57, 15A99 show english descriptions
Vector spaces, linear dependence, rank Linear transformations, semilinear transformations Other types of matrices (Hermitian, skewHermitian, etc.) Miscellaneous topics
15A03  Vector spaces, linear dependence, rank 15A04  Linear transformations, semilinear transformations 15A57  Other types of matrices (Hermitian, skewHermitian, etc.) 15A99  Miscellaneous topics
