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Gershgorin Circle Theorem

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 complex square 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).

Eigenvalue

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References

Bell, H. E. "Gerschgorin's Theorem and the Zeros of Polynomials." Amer. Math. Monthly 72, 292-295, 1965.Brualdi, R. A. and Mellendorf, S. "Regions in the Complex Plane Containing the Eigenvalues of a Matrix." Amer. Math. Monthly 101, 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. Nauk 7, 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. Monthly 56, 672-676, 1949.Varga, R. S. Geršgorin and His Circles. Berlin: Springer-Verlag, 2004.

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Gershgorin Circle Theorem

Cite this as:

Weisstein, Eric W. "Gershgorin Circle Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GershgorinCircleTheorem.html