On Classification and Geometric Characterizations of Ensembled $2\times2$ Pseudo Hermitian and PT-Symmetric Matrices

Abstract: Non-Hermitian matrices $H\in M_2(\mathbb{C})$ satisfying the relation $ H^{\dag}G = GH $, for invertible and singular Hermitian matrices $G$ have been studied. The matrices $H$ corresponding to invertible $G$ are known in the literature as G-pseudo Hermitian matrices. We label the matrices corresponding to the singular $G_s$ as $G_s$-pseudo Hermitian. We have proved that all $ 2\times 2$ $G$-pseudo Hermitian matrices are PT-symmetric. For a given $G$ ($G_s$), all $G$ ($G_s$)-pseudo-Hermitian $H\in M_2(\mathbb{C})$ are found to be expressed as a linear variety. It is further found that for any two Hermitian $G_i,G_j\in M_2(\mathbb{C})$ such that $G_i\neq \lambda G_j$, there always exists exactly one trace less $H\in M_2(\mathbb{C})$ (up to real scaling) which is pseudo-Hermitian with respect to both these $G$ matrices. The set of all $G$- and $G_s$- pseudo-Hermitian matrices has been divided into seven distinct ensembles of matrices and the set of all PT-symmetric matrices in $M_2(\mathbb{C})$ is partitioned into four cells, denoted by $S_1,S_2,S_3$ and $S_4$. The ensembles of trace-less G-pseudo Hermitian matrices are shown to be written as a linear combination of three basis elements from these cells. When $\mathrm{Tr}(G) = 0$, one basis element is from $S_1$ and the other two are from $S_2$. On the other hand, when $\mathrm{Tr}(G)\neq0$, one basis element is from $S_1$ and the other two are from $S_4$. The determinant of such ensembles of trace-less matrices are shown to be quadrics, which could be hyperboloid of two sheets, hyperboloid of one sheet, ellipsoid or quadric cone for invertible $G$, whereas it is two parallel planes or a plane for singular $G_s$. Finally, the set of all the matrices $G\in M_2(\mathbb{C})$, satisfying $H^{\dagger}G = GH$, given a specific $H\in M_2(\mathbb{C})$, are shown to be describable in terms of quadratic variety.