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 n×n matrix A, define

 R_i=sum_(j=1; i!=j)^n|a_(ij)|.

Then each eigenvalue of A is in at least one of the disks


The theorem can be made stronger as follows. Let r be an integer with 1<=r<=n, and let S_j^((r-1)) be the sum of the magnitudes of the r-1 largest off-diagonal elements in column j. Then each eigenvalue of A is either in one of the disks


or in one of the regions

 {z:sum_(i in P)|z-a_(ii)|<=sum_(i in P)R_i},

where P is any subset of {1,2,...,n} such that |P|=r (Brualdi and Mellendorf 1994).

See also


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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.

Referenced on Wolfram|Alpha

Gershgorin Circle Theorem

Cite this as:

Weisstein, Eric W. "Gershgorin Circle Theorem." From MathWorld--A Wolfram Web Resource.

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