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).
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