From Gershgorin Discs to Cassini Ovals in a Weyl Matrix

This is a working note attached to an unpublished project. It explains the mathematical lens; it is not a claim of a finished result.

For a matrix $A=(a_{ij})$, Gershgorin places every eigenvalue in at least one disc

$$G_i=\left\{z\in\mathbb C:\lvert z-a_{ii}\rvert\leq \sum_{j\ne i}\lvert a_{ij}\rvert\right\}.$$

The estimate is inexpensive and robust, but it treats each row separately. Brauer’s refinement couples two rows: every eigenvalue lies in a Cassini region

$$\lvert z-a_{ii}\rvert\,\lvert z-a_{jj}\rvert\leq R_iR_j, \qquad i\ne j,$$

where $R_i=\sum_{k\ne i}\lvert a_{ik}\rvert$. That pairwise product retains information discarded by a union of independent discs.

The Weyl tensor, written in Newman–Penrose variables, produces a small structured complex matrix whose eigenvalue degeneracies encode the Petrov classification. This makes localisation more than a generic numerical bound: the shape and overlap of the regions can be compared with algebraically special limits.

The next checks are concrete: verify the matrix convention from the self-dual bivector construction, compute the regions for canonical Petrov types, test tetrad and scale dependence, compare with exact spectra, and identify counterexamples before assigning an invariant or physical interpretation.