摘要：In(relativistic)electronic structure methods,the quaternion matrix eigenvalue problem and the linear response(Bethe-Salpeter)eigenvalue problem for excitation energies are two frequently encoun-tered structured eigenvalue *** the former problem was thoroughly studied,the later problem in its most general form,namely,the complex case without assuming the positive definiteness of the electronic Hessian,was not fully *** view of their very similar mathematical structures,we examined these two problems from a unified point of *** showed that the identification of Lie group structures for their eigenvectors provides a framework to design diagonalization algorithms as well as numerical optimizations techniques on the corresponding *** using the same reduction algorithm for the quaternion matrix eigenvalue problem,we provided a necessary and sufficient condition to characterize the different scenarios,where the eigenvalues of the original linear response eigenvalue problem are real,purely imaginary,or *** result can be viewed as a natural generalization of the well-known condition for the real matrix case.