1. CJM 2009 (vol 62 pp. 109)
|Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues |
Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$. All possible inertia values, ranks, and multiple eigenvalues of $A + B$ are determined. Extension of the results to the sum of $k$ matrices with $k > 2$ and connections of the results to other subjects such as algebraic combinatorics are also discussed.
Keywords:complex Hermitian matrices, real symmetric matrices, inertia, rank, multiple eigenvalues
2. 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