 Complementary row and column Gershgorin isolation

A matrix and its transpose have the same characteristic polynomial, allowing two independent Gershgorin eigenvalue counting arguments for one spectrum. If isolated row discs certify simple eigenvalues near one group of centres and isolated column discs certify the remaining ones, pairwise separation of all the certified small discs establishes a simple spectrum even if neither full family has pairwise disjoint discs.

For example, let $n=2m$, the diagonal centres be $1,\dots ,n$, and all adjacent off-diagonal entries have magnitude strictly below $1/4$. Add a corner entry $a_{n1}=c$. The row radii are below $1/4,1/2,\dots ,1/2,|c|+1/4$; the column radii reverse the two endpoint bounds. If $|c|\le m-3/4$, rows isolate centres $1,\dots ,m$ and columns isolate centres $m+1,\dots ,n$. The nearest potentially enlarged-disc contact has centre separation $m$, while the radius sum is strictly less than $|c|+3/4\le m$. Each of the $n$ certified small discs contains one simple eigenvalue. For real entries these eigenvalues are real, since a nonreal conjugate pair would put two roots in the same conjugation-invariant isolated disc.

This is a sufficient uniform certificate. Overlap beyond this bound does not prove repeated eigenvalues; individual matrices can have simple real spectra well beyond a Gershgorin disc separation bound.