ordering rows and columns PMP−1, where P is a permutation matrix.) Gershgorin Circle Theorem. Also called the Gershgorin Disk Theorem. Theorem statement. Does every disk have its own eigenvalue? NO but. Theorem. Let A = [aij ] be an n × n complex matrix. If Di1,Di2,,Dik are Gershgorin discs of A that are. has negative determinant, and hence has a negative eigenvalue. Cauchy’s interlacing theorem implies that A too has a negative eigenvalue.

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The Gershgorin circle theorem where “Gershgorin” is sometimes also spelled “Gersgorin” or “Gerschgorin” identifies a region corcle the complex plane that contains all the eigenvalues of a complex square matrix.

For an matrixdefine. Then each eigenvalue of is in at least one of the disks. The theorem can be made stronger as follows.


Let be an integer withand 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.

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Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: Mon Dec 31 Gershgorin Circles Chris Maes. Contact the MathWorld Team.

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