In (relativistic) electronic structure methods, the quaternion matrix eigenvalue problem and the linear response (Bethe-Salpeter) eigenvalue problem for excitation energies are two frequently encountered structured eigenvalue problems. While 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, is not fully understood.
In view of their very similar mathematical structures, we examined these two problems from a unified point of view. We 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 manifolds.
By 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 complex. The result can be viewed as a natural generalization of the well-known condition for the real matrix case.