Gershgorin Circle Theorem

Visualize eigenvalue localization discs and diagonal dominance on the complex plane.

Gershgorin Circle Theorem: Bound the Spectrum Without Solving Polynomials

Concept Overview

Computing the exact eigenvalues of an n × n matrix requires finding roots of an n-th degree characteristic polynomial, which has no closed-form formula for n ≥ 5 by the Abel-Ruffini theorem. The Gershgorin Circle Theorem, published by Soviet mathematician Semyon Aranovich Gershgorin in 1931, provides geometric circular bounds on the complex plane directly from row and column sums of the matrix entries, requiring zero polynomial factorizations.

Mathematical Definition

Let A = (a_ij) ∈ &Copf;^{n × n} be a square matrix. For each row index i ∈ {1, …, n}, the i-th Gershgorin disc D_i is a closed disc in the complex plane centered at the diagonal entry a_ii with radius equal to the sum of absolute values of the off-diagonal entries in that row:

Center: c_i = a_ii

Radius: R_i = Σ_{j ≠ i} |a_ij|

Disc: D_i = { z ∈ &Copf; : |z − a_ii| ≤ R_i }

First Theorem (Eigenvalue Inclusion): Every eigenvalue λ of A lies in at least one Gershgorin disc:

σ(A) ⊆ ∪_i=1ⁿ D_i

Second Theorem (Disjoint Components): If a connected union of k Gershgorin discs is disjoint from the remaining n − k discs, then that connected component contains exactly k eigenvalues of A (counting algebraic multiplicities). In particular, an isolated disc contains exactly one eigenvalue.

Worked Example: Isolating Real and Clustered Eigenvalues

Consider the symmetric 3 × 3 matrix:

A = [
  [ 4, 1, 0],
  [ 1, -2, 1],
  [ 0, 1, 3]
]

Calculate row centers and radii independently:

Row 1: Center c₁ = 4, Radius R₁ = |1| + |0| = 1. Real interval: [4 − 1, 4 + 1] = [3, 5].
Row 2: Center c₂ = −2, Radius R₂ = |1| + |1| = 2. Real interval: [−2 − 2, −2 + 2] = [−4, 0].
Row 3: Center c₃ = 3, Radius R₃ = |0| + |1| = 1. Real interval: [3 − 1, 3 + 1] = [2, 4].

Gershgorin disc intervals and eigenvalue localization for matrix A
Disc	Center a_ii	Radius R_i	Complex Disc Boundary	Real Interval	Component
D₁	4	1	\|z − 4\| ≤ 1	[3, 5]	Component 2
D₂	−2	2	\|z + 2\| ≤ 2	[−4, 0]	Component 1 (Isolated)
D₃	3	1	\|z − 3\| ≤ 1	[2, 4]	Component 2

Notice that D₁ and D₃ overlap since [3, 5] ∩ [2, 4] = [3, 4] ≠ ∅. Their union forms Component 2 spanning [2, 5]. Meanwhile, D₂ has interval [−4, 0], which is completely disjoint from [2, 5] since 0 < 2.

By Gershgorin's second theorem:

Component 1 (D₂) is an isolated disc (k = 1), so it contains exactly 1 eigenvalue in [−4, 0].
Component 2 (D₁ ∪ D₃) is a union of k = 2 discs, containing exactly 2 eigenvalues in [2, 5].

Direct calculation of the characteristic polynomial yields p(λ) = λ³ − 5λ² − 4λ + 31 = 0, whose numerical roots are:

λ₁ ≈ 4.1871 ∈ [3, 5] ⊆ [2, 5]
λ₂ ≈ −2.3447 ∈ [−4, 0]
λ₃ ≈ 3.1576 ∈ [2, 4] ⊆ [2, 5]

Every eigenvalue lies inside its predicted component with zero error.

Run and Experiment with Gershgorin Discs

Open the Gershgorin Circle Theorem interactive explorer to test how perturbation, coupling, orientation, and similarity scaling alter the circular bounds:

Select Matrix Preset: strictly-dominant. The discs are centered at 4, −3, and 5 with radii 1, 2, 1. Because |a_ii| > R_i for all i, 0 is excluded from all discs, verifying strict diagonal dominance and invertibility.
Drag Off-Diagonal Coupling to 0. Notice that all radii collapse to R_i = 0. The circular discs shrink into point markers directly on the diagonal entries, which coincide exactly with the matrix eigenvalues {4, −3, 5}.
Select Matrix Preset: disjoint-discs. Observe the isolated disc 2 at center −2 with radius 2, completely separated from discs 1 and 3. Check that the disjoint components counter displays 2.
Toggle Disc Orientation to column. The discs now compute radii from column sums C_j = Σ_{i ≠ j} |a_ij| of A (row sums of A^T). Because eigenvalues of A and A^T are identical, the intersection of row and column discs tightens eigenvalue localization.
Adjust Similarity Scale Factor s from 1.0 to 2.0. This applies the similarity transformation D⁻¹AD with D = diag(s, 1, 1). Observe that disc 1's radius shrinks from 1.0 to 0.5 because its off-diagonal entries are divided by s, while discs 2 and 3 expand because their first columns are multiplied by s. The true eigenvalues remain completely stationary!

Mathematical Assumptions and Limitations

While Gershgorin discs provide instantaneous bounds without solving algebraic equations, their practical application involves several crucial nuances:

Individual discs need not contain an eigenvalue: When two or more discs overlap into a union of k > 1 discs, the theorem guarantees that the union contains k eigenvalues. It does not guarantee that each individual overlapping disc contains at least one eigenvalue. An individual disc in a cluster may be entirely devoid of eigenvalues.
Discs are conservative bounds: For non-normal matrices or matrices with large off-diagonal entries, Gershgorin discs can be significantly larger than the actual convex hull of eigenvalues.
Invertibility converse does not hold: If a matrix is strictly diagonally dominant (|a_ii| > R_i for all i), the origin 0 is outside all discs, proving A is invertible. However, if a disc contains 0, the matrix is not necessarily singular; it simply cannot be proven invertible by Gershgorin's test alone.
Similarity transformations: Any diagonal similarity D⁻¹AD preserves eigenvalues while reshaping discs. The true spectrum satisfies σ(A) ⊆ ∩_{D > 0} ∪_i=1ⁿ D_i(D⁻¹AD), known as the minimal Gershgorin set.

Primary References and Scientific Sources

The mathematical principles, localization theorems, and diagonal scaling invariants presented in this module are drawn from foundational literature in numerical linear algebra and matrix analysis:

Weisstein, Eric W. "Gershgorin Circle Theorem." From MathWorld--A Wolfram Web Resource — Explains the formal inclusion theorem, disjoint union theorem, and column-disc intersection properties.
Bindel, David. "Gershgorin Theory and Perturbation of Eigenvalues", Cornell University CS6210 Lecture Notes (2016) — Rigorous treatment of diagonal similarity scaling, continuity of eigenvalues, and Gershgorin disc isolation.

Gershgorin Circle Theorem