2024/10/24 by Stalin Abraham, Abraham, Stalin, Ameeya A. Bhagwat +1
Computer Science · Physics and Astronomy · #Advanced Mathematical Theories and Applications #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.2410.18530
openalex publication_date 2024/10/24 · openalex created_date 2024/11/13 · openalex updated_date 2026/07/28
Non-Hermitian matrices H∈ M2(ℂ) satisfying the relation H^†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 Gs as Gs-pseudo Hermitian. We have proved that all 2× 2 G-pseudo Hermitian matrices are PT-symmetric. For a given G (Gs), all G (Gs)-pseudo-Hermitian H∈ M2(ℂ) are found to be expressed as a linear variety. It is further found that for any two Hermitian Gi,Gj∈ M2(ℂ) such that Gi≠ λGj, there always exists exactly one trace less H∈ M2(ℂ) (up to real scaling) which is pseudo-Hermitian with respect to both these G matrices. The set of all G- and Gs- pseudo-Hermitian matrices has been divided into seven distinct ensembles of matrices and the set of all PT-symmetric matrices in M2(ℂ) is partitioned into four cells, denoted by S1,S2,S3 and S4. 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 Tr(G) = 0, one basis element is from S1 and the other two are from S2. On the other hand, when Tr(G)≠0, one basis element is from S1 and the other two are from S4. 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 Gs. Finally, the set of all the matrices G∈ M2(ℂ), satisfying H†G = GH, given a specific H∈ M2(ℂ), are shown to be describable in terms of quadratic variety.