Gershgorin disc ($D_i(A)=\{z\in \mathbb{C}:|z-a_{ii}|\le r_i\}$)

A row Gershgorin disc of a square matrix $A=(a_{ij})$ is the closed disk in the complex plane with centre $a_{ii}$ and radius $r_i={\sum }_{j\ne i}|a_{ij}|$. A zero radius gives a single point. The Gershgorin disc theorem places every eigenvalue in the union of these disks. For a Hermitian matrix, the real intervals cut out by the Gershgorin discs bound the eigenvalues, including energy levels when the matrix represents a Hamiltonian operator.

Applying the Gershgorin circle theorem to the transpose $A^T$ gives column Gershgorin discs with the same centres and radii $c_i={\sum }_{j\ne i}|a_{ji}|$. Since $A$ and $A^T$ have the same characteristic polynomial, every eigenvalue lies in the intersection of the row-disc union and the column-disc union. This is an intersection of two unions; one cannot generally replace each row radius by $\min (r_i,c_i)$. An isolated row Gershgorin disc contains exactly one eigenvalue by Gershgorin eigenvalue counting.^[ref]