Gersgorin Circle Theorem

Gives a region in the Complex Plane containing all the Eigenvalues of a Complex Square Matrix. Let

$$\vert x_k\vert = \mathop{\rm max}\{\vert x_i\vert:1\leq i\leq n\}>0$$ (1)

and define

$$R_i=\sum_{\scriptstyle i=1 \atop \scriptstyle j\not = i}^n \vert a_{ij}\vert.$$ (2)

Then each Eigenvalue of the Matrix A of order $n$ is in at least one of the disks

$$\{z:\vert z-a_{ii}\vert\leq R_i\}.$$ (3)

The theorem can be made stronger as follows. Let $r$ be an Integer with $1\leq r\leq n$, then each Eigenvalue of A is either in one of the disks $\Gamma_1$

$$\{z:\vert z-a_{jj}\vert\leq S_j^{(r-1)}\},$$ (4)

or in one of the regions

$$\left\{{z:\sum_{i=1}^r \vert z-a_{ii}\vert\leq \sum_{i=1}^r R_i}\right\},$$ (5)

where $S_j^{(r-1)}$ is the sum of magnitudes of the $r-1$ largest off-diagonal elements in column $j$.

References

Brualdi, R. A. and Mellendorf, S. ``Regions in the Complex Plane Containing the Eigenvalues of a Matrix.'' Amer. Math. Monthly 101, 975-985, 1994.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1120-1121, 1979.

Taussky-Todd, O. ``A Recurring Theorem on Determinants.'' Amer. Math. Monthly 56, 672-676, 1949.