The Pseudo-Hermitian Condition

The decomposition of pseudo-Hermitian matrices in terms of their eigenvalue and their eigenvectors is discussed. Firstly, biorthogonality is considered as the usual tool to perform matrix decomposition of non-Hermitian case. However, also it is shown that the decomposition can be obtained by considering the metric. Next, the Dyson’s trick to map a pseudo-Hermitian matrix into a Hermitian one is described. Finally, the special case of ensembles of tridiagonal matrices is discussed.