Pseudo-orthogonal diagonalization for linear response eigenvalue problems

We present a pseudo-QR algorithm that solves the linear response eigenvalue problem ℋ x = λx. ℋ is known to be Π-symmetric with respect to T = diag {J, −J}, where J(i, i) = ±1 and J(i, j) = 0 when i ≠ j. Moreover, y^∗Tx = 0 if λ ≠ γ̄ for eigenpairs (λ, x) and (γ, y). The employed algorithm was designed for solving the eigenvalue problem Qv = σv for pseudo-orthogonal matrix Q such that Q′TQ = T. Although ℋ is not orthogonal with respect to T, the pseudo-QR algorithm is able to transform ℋ into a quasi-diagonal matrix with diagonal blocks of size 2 × 2 using J-orthogonal transforms. This guarantees the pair-wise appearance of the eigenvalues λ and −λ of ℋ.