1. CJM 2008 (vol 60 pp. 1050)
||Adjacency Preserving Maps on Hermitian Matrices |
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.
Keywords:rank, adjacency preserving map, hermitian matrix, geometry of matrices
Categories:15A03, 15A04, 15A57, 15A99
2. CJM 1997 (vol 49 pp. 840)
||Non-Hermitian solutions of algebraic Riccati equation |
Non-hermitian solutions of algebraic matrix Riccati
equations (of the continuous and discrete types) are studied. Existence
is proved of non-hermitian solutions with given upper bounds of the
ranks of the skew-hermitian parts, under the sign controllability
Categories:15A99, 15A63, 93C60