Physical and unphysical solutions of random-phase approximation equation

Properties of solutions of the RPA equation is reanalyzed mathematically, which is defined as a generalized eigenvalue problem of the stability matrix $\mathsf{S}$ with the norm matrix $\mathsf{N}=\mathrm{diag.}(1,-1)$. As well as physical solutions, unphysical solutions are examined in detail, with taking the possibility of Jordan blocks of the matrix $\mathsf{N\,S}$ into consideration. Two types of duality of eigenvectors and basis vectors of the Jordan blocks are pointed out and explored, which disclose many basic properties of the RPA solutions.