Adjacency Preserving Maps on Hermitian Matrices
Printed: Oct 2008
Peter \v Semrl
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.
rank, adjacency preserving map, hermitian matrix, geometry of matrices
15A03 - Vector spaces, linear dependence, rank
15A04 - Linear transformations, semilinear transformations
15A57 - Other types of matrices (Hermitian, skew-Hermitian, etc.)
15A99 - Miscellaneous topics