Introduction

The eigenvalues of a diagonal matrix are very easy to locate, and the eigenvalues of a matrix are continuous functions of the entries, so it is natural to ask whether one can say anything useful about the eigenvalues of a matrix whose off-diagonal elements are “small” relative to the main diagonal entries. Such matrices do arise in practice; large systems of linear equations resulting from numerical discretization of boundary value problems for elliptic partial differential equations can be of this form.

In some differential equations problems involving the long-term stability of an oscillating system, one is sometimes interested in showing that the eigenvalues {λi} of a matrix all lie in the left half-plane, that is, that Re(λi)<0. And sometimes in statistics or numerical analysis one needs to show that a Hermitian matrix is positive definite, that is, that all λi > 0.

Sometimes one wants to locate the eigenvalues of a matrix in a bounded set that is easily characterized. We know that all the eigenvalues of a matrix A are located in a disc in the complex plane centered at the origin and having radius ∥A∥, where ∥·∥ is any matrix norm. But can one do better than this by more precisely locating regions that must either include or exclude the eigenvalues? We shall see that one can.