= Gershgorin disc
{c}
{title2=$D_i(A)=\{z\in\mathbb C:|z-a_{ii}|\leq r_i\}$}
= Gershgorin disk
{c}
{synonym}
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>.https://www.d.umn.edu/~mhampton/NA/Eigenstuff/[Marshall Hampton, Introduction to Numerical Methods, Eigenstuff: Gershgorin circle theorem]{ref}
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