Jacobi method for generalised hermitian eigenproblems

For conservative systems, the governing equations in matrix form are self-adjoint in the spatial and temporal coordinates. The imaginary parts come from the odd orders of the time or spacederivatives. The homogeneous solution of the resulting set of ordinary differential equations requires the solution of thegeneralised Hermitian eigenproblem [A] {x} =λ[B] {x}, where both [A] and {B} are Hermitian. If [A] or [B] is positive definite, thegeneralised Hermitian eigenproblem can be reduced to the standard Hermitian eigenproblem involving one Hermitian matrix only. Otherwise, the reduction is not possible and the left eigenvector is not the complex conjugate of the right eigenvector in contrast to the standard Hermitian eigenproblem. A new method of rotation is proposed to solve the generalised Hermitian eigenproblem without the requirement of definiteness. Necessary condition is given for the eigenproblem to have real eigenvalues. The proposed method reduces both matrices to diagonal form and is able to provide all real eigenvalues.