As I understand it there are ensembles which are measures on spaces of unitary matrices. I am also interested in the statistical properties of the eigenvalues of a antihermitian matrix whose entries are Gaussian random variables.
There is a small amount of information on this in the book by Mehta where he obtains some analytic results for them. However, in this book he states that the eigenvalues of an antisymmetric Hermitian matrix always come in pairs $+x_i$ and $-x_i$, where $x_i$ is real. I am slightly confused by this as I thought that the eigenvalues of a skew-Hermitian matrix had to be pure imaginary. Is an antisymmetric Hermitian matrix not the same as a antihermitian matrix?
Best Answer
A skew Hermitian matrix is just $i$ times a Hermitian matrix, so its matrix eigenvalue statistics are (up to the trivial factor of $i$) identical to that of a Hermitian matrix. Any Hermitian matrix can be writen as $H= X+iY$, where $X$ is symmetric with real entries and $Y$ is skew-symmetric with real entries. A skew-symmetric Hermitian matrix is one such that $X=0$. It is thus very different from a skew Hermitian matrix and indeed its eigenvalues come in $\pm \lambda_i $ pairs.