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The Gershgorin circle theorem locates every eigenvalue of a square matrix using only its entries. A diagonal entry supplies a centre in the complex plane, and the sum of the off-diagonal absolute values in its row supplies a radius. Small off-diagonal couplings therefore confine the eigenvalues near the diagonal entries. This turns a global eigenvalue problem into a picture built one row at a time.

Precisely, let $A=(a_{ij})\in {\mathbb{C}}^{n\times n}$, with $n\ge 1$, and define the closed Gershgorin discsEvery eigenvalue $\lambda$ of $A$ satisfies $\lambda \in {\bigcup }_i{D}_i(A)$, including eigenvalues on the boundaries. No symmetry or diagonalizability is required. Despite the name, the theorem uses filled disks, not just their circular boundaries.^[ref]

The short reason is the triangle inequality. For a nonzero eigenvector $x$ with $Ax=\lambda x$, choose $i$ with $|x_i|={\max }_j|x_j|>0$. The $i$th row givesThus the largest eigenvector coordinate identifies a Gershgorin disc containing $\lambda$, even though that eigenvector need never be computed to use the bound.

This is useful when a full eigenvalue computation would be expensive: forming the Gershgorin discs needs one pass over the nonzero entries of a sparse matrix. If $0$ lies outside every Gershgorin disc, then $A$ is an invertible matrix, since zero is not an eigenvalue. The condition is exactly $|a_{ii}|>r_i$ in every row, namely strict diagonal dominance. This connects diagonal dominance to solvability of a linear system; the inverse infinity-norm bound from diagonal dominance adds a quantitative bound. It is a sufficient test: failure to exclude zero does not prove singularity.

For a concrete symmetric matrix, takeThe three Gershgorin discs have centres $2,5,9$ and are pairwise disjoint. The finite-dimensional spectral theorem makes the eigenvalues real, so their possible locations reduce to the intervals $[1.7,2.3]$, $[4.5,5.5]$, and $[8.6,9.4]$. Gershgorin eigenvalue counting gives exactly one eigenvalue in each. Independently, the characteristic polynomial iswhose numerical polynomial roots are $1.985811$, $4.989942$, and $9.024247$, rounded to six decimal places. All are positive, as Gershgorin bounds for symmetric matrices already guarantee without finding those polynomial roots.

S. Gerschgorin published the result in 1931 in Über die Abgrenzung der Eigenwerte einer Matrix, in the Bulletin de l'Académie des Sciences de l'URSS, issue 6, pages 749–754. The original paper's §2 states both eigenvalue inclusion and counting in separated groups of Gershgorin discs; §3 already discusses column bounds and transformations preserving the eigenvalues. Thus both the localization picture and ways to sharpen it were part of the original development.^[ref]

A symmetric matrix obtained from a conservative five-point discretization of a diffusion equation has row centre $-s_i/h^2$ and radius at most $s_i/h^2$, where $h>0$ is the grid spacing and $s_i$ is the sum of its four nonnegative edge coefficients. Here the matrix represents the diffusion generator, with nonnegative off-diagonal entries; boundary elimination may reduce the row radius. If every coefficient is at most $a_{\max }>0$, then $s_i\le 4a_{\max }$, and the Gershgorin bounds for symmetric matrices place every eigenvalue in $[-8a_{\max }/h^2,0]$.

For the Forward Euler method, an eigenvalue $\lambda \le 0$ has amplification factor $1+\Delta t\lambda$. Requiring $|1+\Delta t\lambda |\le 1$ for every such eigenvalue gives the sufficient absolute stability condition $\Delta t/h^2\le 1/(4a_{\max })$. Symmetry makes this also prevent growth of the Euclidean norm. For $a_{\max }=2$ and $h=0.1$, the bound is $\lambda \in [-1600,0]$ and $\Delta t\le 0.00125$. At that step size the interval maps to amplification factors in $[-1,1]$. This illustrates why cheap Gershgorin disc bounds can select a stable time step before any eigenvalues are computed.