For row of a matrix, the Gershgorin disc has centre and radius . The Gershgorin disc theorem places every eigenvalue in the union of these discs. For a Hermitian matrix the real intervals cut out by the discs bound energies; an isolated disc contains exactly one eigenvalue.
A symmetric conservative five-point diffusion matrix has row center and radius at most , where is the sum of its four nonnegative edge coefficients. If every coefficient is at most , its eigenvalues lie in . Forward Euler is therefore stable when .
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The Gershgorin Circle Theorem is a result in linear algebra that provides a method for locating the eigenvalues of a square matrix. It is particularly useful when analyzing the spectral properties of a matrix without explicitly calculating its eigenvalues. The theorem states that for any \( n \times n \) complex matrix \( A = [a_{ij}] \), the eigenvalues of \( A \) lie within certain circles in the complex plane defined by the rows of the matrix.