Sturm theorem for quaterion Hermitian eigenproblems

Abstract

An important problem in quantum mechanics involving time reversal symmetry and inversion symmetry is the computation of a quaternion Hermitian matrix [Q] ϵH^{n x n} which is equal to its conjugate transpose [Q]^T=[Q] . Since [Q] is represented by a 2n × 2n complex matrix [C] ϵC^{2n x 2n} which is in turn represented by a 4n × 4n real matrix [R] ϵR^{4n x 4n} , the eigensolution of [Q] is equivalent to that of [C] or [R] . However, the Hermitian property will be lost if the original quaternion matrix is solved by means of its real or complex representations. In this paper, we solve the quaternion Hermitian matrix by means of quaternion arithmetics and preserve all the advantage of the band structure and the Hermitian property. We introduce a generalized Sturm theorem to include quaternion Hermitian matrices which treats complex Hermitian matrices and symmetric real matrices as special cases.