Brauer's Theorem

If, in the Gersgorin Circle Theorem for a given $m$,

$$\vert a_{jj}-a_{mm}\vert>\Lambda_j+\Lambda_m$$

for all $j\not=m$, then exactly one Eigenvalue of ${\hbox{\sf A}}$ lies in the Disk $\Gamma_m$.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1121, 1979.