The Gershgorin circle theorem (where "Gershgorin" is sometimes also spelled "Gersgorin" or "Gerschgorin") identifies a region in the complex
plane that contains all the eigenvalues of a complexsquare matrix.
For an matrix, define
(1)
Then each eigenvalue of is in at least one of the disks
(2)
The theorem can be made stronger as follows. Let be an integer with , and let be the sum of the magnitudes of the largest off-diagonal elements in column . Then each eigenvalue of
is either in one of the disks
(3)
or in one of the regions
(4)
where
is any subset of such that (Brualdi and Mellendorf 1994).
Bell, H. E. "Gerschgorin's Theorem and the Zeros of Polynomials." Amer. Math. Monthly72, 292-295, 1965.Brualdi,
R. A. and Mellendorf, S. "Regions in the Complex Plane Containing the Eigenvalues
of a Matrix." Amer. Math. Monthly101, 975-985, 1994.Feingold,
D. G. and Varga, R. S. "Block Diagonally Dominant Matrices and Generalizations
of the Gerschgorin Circle Theorem." Pacific J. Math.12, 1241-1250,
1962.Gerschgorin, S. "Über die Abgrenzung der Eigenwerte einer
Matrix." Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk7, 749-754,
1931.Gradshteyn, I. S. and Ryzhik, I. M. Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
pp. 1073-1074, 2000.Piziak, R. and Turner, D. "Exploring Gerschgorin
Circles and Cassini Ovals." Mathematica Educ. and Res.3, 13-21,
1994.Scott, D. S. "On the Accuracy of the Gerschgorin Circle
Theorem for Bounding the Spread of a Real Symmetric Matrix." Lin. Algebra
Appl.65, 147-155, 1985.Taussky-Todd, O. "A Recurring
Theorem on Determinants." Amer. Math. Monthly56, 672-676, 1949.Varga,
R. S. Geršgorin
and His Circles. Berlin: Springer-Verlag, 2004.
matrix 
, define
is in at least one of the disks
be an integer with 
, and let 
be the sum of the magnitudes of the 
largest off-diagonal elements in column 
. Then each eigenvalue of 
is either in one of the disks
is any subset of 
such that 
(Brualdi and Mellendorf 1994).
