Inclusion regions for matrix eigenvalues Original Research Article

We review Lehmann’s inclusion bounds and provide extensions to general (non-normal) matrices. Each inclusion region has a diameter related to the singular values of a restriction of the matrix to a subspace and dependent on either an eigenvector condition number or the departure of the matrix from normality. The inclusion regions are optimal for normal matrices. Similar considerations lead to inclusion bounds based on relative distances expressed analogously in terms of appropriately defined generalized singular values.