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Generalize the Cullum–Willoughby residual–characteristic‑polynomial identity to block Lanczos

Derive a version of the Cullum–Willoughby identity relating the scalar residual bound to characteristic polynomials for oblique block Lanczos with block size r > 1, enabling a direct residual‑based eigenvalue filtering criterion analogous to the scalar case.

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Background

In scalar Lanczos, the Cullum–Willoughby test leverages an identity that ties the residual bound to characteristic polynomials of minors, providing a principled way to flag spurious eigenvalues. The authors confirm the identity’s validity for oblique scalar Lanczos and use it to motivate filtering.

They explicitly note that this identity is only established for the scalar (r=1) case and that a simple generalization valid for block Lanczos with r>1 is not known, motivating the development of an analogous identity for block Lanczos.

References

As noted in Eq.~eq:CW_resid, this identity is only valid for scalar Lanczos ($r=1$) and we are not aware of a simple generalization that is valid for block Lanczos with $r>1$."}]}

eq:CW_resid:

Bk(m)=∏j=2m+1∣βj∣2a~m(λk(m))am−1′(λk(m)),(r=1),B_k^{(m)} = \frac{\prod_{j=2}^{m+1} |\beta_j|^2}{\widetilde{a}_m\left(\lambda_k^{(m)}\right) a_{m-1}'\left(\lambda_k^{(m)}\right) }, \hspace{15pt} (r=1),

— Block Lanczos for lattice QCD spectroscopy and matrix elements (2412.04444 - Hackett et al., 5 Dec 2024) in Section 5.2 (The Cullum‑Willoughby test)