Index of a matrix, complex logarithms, and multidimensional Fresnel integrals

TL;DR

This paper critically examines the calculation of the eigenvalue index of a real symmetric matrix using complex logarithms, highlighting potential pitfalls and proposing improvements with multidimensional Fresnel integrals, supported by examples.

Contribution

It identifies issues with a common formula for eigenvalue counting and introduces an improved approach using multidimensional Fresnel integrals, though uniqueness of solutions remains uncertain.

Findings

The standard formula can fail due to branch-cut issues.

Multidimensional Fresnel integrals improve the formula.

Solution uniqueness for the improved formula is not guaranteed.

Abstract

We critically discuss the problem of finding the $\lambda$-index $\mathcal{N}(\lambda)\in [0,1,\ldots,N]$ of a real symmetric matrix $\mathbf{M}$, defined as the number of eigenvalues smaller than $\lambda$, using the entries of $\mathbf{M}$ as only input. We show that a widely used formula $$ \mathcal{N}(\lambda)=\lim_{\epsilon\to 0^+}\frac{1}{2\pi \mathrm{i}}\left[\log\det(\mathbf{M}-\lambda+\mathrm{i}\epsilon)-\log\det(\mathbf{M}-\lambda-\mathrm{i}\epsilon)\right] $$ based on the branch-cut structure of the complex logarithm should be handled with care, as it generically fails to produce the correct result if the same branch is chosen for the two logarithms. We improve the formula using multidimensional Fresnel integrals, showing that even the new version provides at most a self-consistency equation for $\mathcal{N}(\lambda)$, whose solution is not guaranteed to be unique. Our results are corroborated by explicit examples and numerical evaluations.