 Solution

Put $n=2m$. With $b=2$, the row Gershgorin discs have centres $1,2,\dots ,n$ and radiiApplying the theorem to the transpose gives column Gershgorin discs with radiiThe enlarged disc moves from the last centre to the first centre. All small discs in either family are mutually disjoint, because adjacent centres are one unit apart and their radii have sum strictly below one.

For the row family, the small disc at any $i\le m$ is also disjoint from the enlarged disc at $n$ ifFor the column family, every small disc at $i\ge m+1$ is disjoint from the enlarged disc at $1$ by the same estimate, since $i-1\ge m$. Thus the Gershgorin eigenvalue counting theorem certifies one simple eigenvalue near each of $1,\dots ,m$ from rows, and near each of $m+1,\dots ,n$ from columns. The certified small discs from the two families are themselves disjoint, so these are $n$ different eigenvalues.

The largest nonnegative $c$ certified by these uniform row-and-column separation bounds isMore generally the argument works for $|c|\le c_{\mathrm{G}}$. The endpoint is included because all actual entry bounds, and hence all actual radius estimates above, are strict. For $n=8$, $c_{\mathrm{G}}=13/4$: row discs isolate centres $1,2,3,4$, and column discs isolate centres $5,6,7,8$.

For $c>c_{\mathrm{G}}$, taking the adjacent entries sufficiently close to $1/4$ makes the enlarged row disc meet the small disc at $m$, and the enlarged column disc meet the small disc at $m+1$. Hence this particular uniform disc-separation certificate stops there. This is a Gershgorin guarantee, not a necessary spectral threshold: failure of disc separation does not prove repeated or nonreal eigenvalues for the actual matrix.

Finally, a real matrix has nonreal eigenvalues in complex conjugate pairs. Every certified small disc has a real centre and is invariant under conjugation. A nonreal eigenvalue in it would bring its distinct conjugate into the same disc, contradicting its count of exactly one. Therefore all the certified eigenvalues are real and simple. The two-family argument is Complementary row and column Gershgorin isolation.