The Gershgorin circle theorem, http://en.wikipedia.org/wiki/Gershgorin_circle_theorem, gives bounds on the eigenvalues of a square matrix, and works well for nearly diagonal matrices.
For a triangular matrix, however, the bounds are not useful in general, despite the fact that the eigenvalues are known to be the diagonal elements. Is there a version (or can some helpful person develop a version) of the Gershgorin circle theorem that gives more useful bounds in the nearly triangular case?