## Gersgorin Circle Theorem

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

 (1)

and define
 (2)

Then each Eigenvalue of the Matrix A of order is in at least one of the disks
 (3)

The theorem can be made stronger as follows. Let be an Integer with , then each Eigenvalue of A is either in one of the disks
 (4)

or in one of the regions
 (5)

where is the sum of magnitudes of the largest off-diagonal elements in column .

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.