On the accuracy of the Gerschgorin circle theorem for bounding the spread of a real symmetric matrix

Abstract

The spread of a matrix with real eigenvalues is the difference between its largest and smallest eigenvalues. The Gerschgorin circle theorem can be used to bound the extreme eigenvalues of the matrix and hence its spread. For nonsymmetric matrices the Gerschgorin bound on the spread may be larger by an arbitrary factor than the actual spread even if the matrix is symmetrizable. This is not true for real symmetric matrices. It is shown that for real symmetric matrices (or complex Hermitian matrices) the ratio between the bound and the spread is bounded by p+1 , where p is the maximum number of off diagonal nonzeros in any row of the matrix. For full matrices this is just n . This bound is not quite sharp for n greater than 2, but examples with ratios of n−1 for all n are given. For banded matrices with m nonzero bands the maximum ratio is bounded by m independent of the size of n. This bound is sharp provided only that n is at least 2 m. For sparse matrices, p may be quite small and the Gerschgorin bound may be surprisingly accurate.