Gershgorin eigenvalue counting ($\#\{\lambda \text{ in }C\}=k\text{with algebraic multiplicity}$)

Let $C$ be a connected component of the union of the row Gershgorin discs of an $n\times n$ matrix. If $C$ comprises $k$ of the indexed Gershgorin discs, then it contains exactly $k$ eigenvalues, counted with algebraic multiplicity. Coincident disks still count separately, once for each row. In particular, a Gershgorin disc disjoint from all the others contains one simple eigenvalue. More generally, any union of $k$ indexed Gershgorin discs disjoint from the remaining $n-k$ contains exactly $k$ eigenvalues, whether or not that union is connected.^[ref]

The separation is essential: overlapping Gershgorin discs do not individually promise one eigenvalue each. Discs that touch at a boundary point belong to the same connected component. For example, the matrix $\left(\begin{array}{ccc}2& 1& 0\\ 0& 2& 0\\ 0& 0& 8\end{array}\right)$ has Gershgorin discs of radii $1,0,0$ with centres $2,2,8$. The component around $2$ comprises two indexed disks and contains the eigenvalue $2$ with algebraic multiplicity two; the isolated point at $8$ contains the remaining eigenvalue.

For intuition, continuously turn on the off-diagonal entries using the diagonal matrix $E=\mathrm{diag}(a_{11},\dots ,a_{nn})$ and $A(t)=E+t(A-E)$, $0\le t\le 1$. At $t=0$ the eigenvalues are the disk centres; at every later time the Gershgorin discs stay inside their final union. The continuity of the eigenvalues as a collection means that no eigenvalue can cross a gap between final components. The count gives a useful certificate that an eigenvalue cluster is complete.^[ref]

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.