Pseudo-Orthogonal Diagonalization for Linear Response Eigenvalue Problems

• Main Authors: Oliveros-Yusiong, Annie Lyn, David, Roden Jason A.
• Format: text
• Published: Archīum Ateneo 2024
• Subjects: Mathematics; Physical Sciences and Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/260; https://doi.org/10.1063/5.0193353
• Institution: Ateneo De Manila University

We present a pseudo-QR algorithm that solves the linear response eigenvalue problem ℋ x = γx. 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 γ ≠ γ¯ for eigenpairs (γ,x) and (γ,y). The employed algorithm was designed for solving the eigenvalue problem Qv = σv for pseudoorthogonal 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 γ and -γ of ℋ.