This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Gerschgorin's circle theorem | |
|---|---|
| Name | Gerschgorin's circle theorem |
| Field | Linear algebra |
| First proved | 1931 |
| Author | Russian mathematician |
Gerschgorin's circle theorem is a fundamental result in linear algebra that gives bounds on the locations of eigenvalues of a complex square matrix by constructing discs in the complex plane. The theorem connects matrix entries to spectral information and is used across numerical analysis, operator theory, and applied mathematics. It is widely cited alongside results by David Hilbert, John von Neumann, Alfred Tarski, Issai Schur, and Richard Courant in classical spectral theory.
For any n×n complex matrix A = (a_{ij}), define for each index i a closed disc in the complex plane centered at a_{ii} with radius R_i = sum_{j≠i} |a_{ij}|. Then every eigenvalue of A lies in the union of these n discs. This compact formulation complements spectral inclusion results by Carl Friedrich Gauss, William Rowan Hamilton, Augustin-Louis Cauchy, Pafnuty Chebyshev, and Émile Picard that relate algebraic roots and matrix spectra. A common corollary states that if a single disc is disjoint from the others, then it contains exactly one eigenvalue, a fact used in perturbation theory developed by Erwin Schroedinger, Paul Dirac, John von Neumann, and Mark Kac.
Elementary proofs use row operations and the triangle inequality as in expositions by Issai Schur and Hermann Weyl, while alternative proofs invoke analytic function theory à la Émile Picard and operator norms related to work by John von Neumann and Stefan Banach. Matrix norm formulations replace entrywise radii by induced norms, echoing approaches in the work of David Hilbert and Felix Hausdorff. Variations include column-based discs, block versions attributable to methods used by Emil Artin and Richard Bellman, and weighted discs introduced in literature influenced by Eugene Wigner and Norbert Wiener. General proofs exploit Gershgorin-type lemmas in finite-dimensional spectral theory alongside interlacing results associated with Friedrich Hund, Hermann Weyl, and Alfred Tarski.
Gerschgorin discs are applied in eigenvalue estimation for matrices arising in numerical methods attributed to John von Neumann, Alan Turing, C. Lanczos, and James H. Wilkinson, and in stability analysis of dynamical systems following frameworks by Henri Poincaré, Andrey Kolmogorov, Lev Pontryagin, and Stephen Smale. In control theory and circuit analysis the theorem integrates with methods developed by Rudolf Kalman, Norbert Wiener, Claude Shannon, and Oliver Heaviside to bound poles of transfer functions. Computational packages inspired by work at Bell Labs, IBM Research, Mathematics Research Center (Madison), and INRIA use Gerschgorin estimates for preconditioning and spectral clustering tasks linked to research by S. Lloyd, Johan Åström, and Michael I. Jordan.
For a diagonally dominant matrix such as those studied by Carl Friedrich Gauss and Adrien-Marie Legendre in numerical linear algebra, the discs are small and provide precise eigenvalue localization comparable to bounds used in algorithms by C. Lanczos and James H. Wilkinson. Consider tridiagonal matrices arising in finite-difference discretizations found in works of Richard Courant and David Hilbert; Gerschgorin discs yield estimates akin to those used in spectral methods developed by S. Chandrasekhar and Kurt Friedrichs. In perturbation examples inspired by Paul Dirac and Eugene Wigner, a rank-one change shifts disc centers and radii, illustrating isolation of eigenvalues similar to phenomena discussed by John von Neumann and Oskar Perron.
Extensions include Ostrowski and Brauer theorems named alongside contributions by Alexander Ostrowski, Heinrich Brauer, Issai Schur, and Hermann Weyl, which refine discs using permutations or pairs of rows and columns; these relate to Perron–Frobenius theory developed by Oskar Perron and Georg Frobenius. Further generalizations involve pseudospectra research by Lloyd N. Trefethen, operator-valued analogues in functional analysis following Stefan Banach and John von Neumann, and inclusion sets derived from Gershgorin ideas used in modern matrix analysis by Roger Horn and Charles R. Johnson. Connections link the theorem to eigenvalue perturbation theory by Thomas Kato, to localization techniques in random matrix theory studied by Eugene Wigner and Freeman Dyson, and to numerical stability results in computational mathematics associated with James H. Wilkinson and Nicholas Higham.
Category:Theorems in linear algebra