= Complementary row and column Gershgorin isolation
{c}
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 $1,\ldots,n$, and all adjacent off-diagonal entries have magnitude strictly below $1/4$. Add a corner entry $a_{n1}=c$. The row radii are below $1/4,1/2,\ldots,1/2,|c|+1/4$; the column radii reverse the two endpoint bounds. If $|c|\leq m-3/4$, rows isolate centres $1,\ldots,m$ and columns isolate centres $m+1,\ldots,n$. The nearest potentially enlarged-disc contact has centre separation $m$, while the radius sum is strictly less than $|c|+3/4\leq m$. 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.
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