SCALED ALMOST DIAGONAL MATRICES WITH MULTIPLE SINGULAR-VALUES

New estimates for relative distances between the singular values and t he moduli of the appropriate diagonal elements of a scaled almost diag onal square matrix are derived. In case of a multiple singular value t he bunds also estimate the structure of the diagonal block associated with that singular value. The bounds are expressed in terms of the off -diagonal elements of an appropriately scaled matrix and of relative g aps between singular values. The new estimates refine the existing one s which are based on the absolute gaps between singular values. They a re especially appropriate for the smallest singular values. For triang ular and essentially triangular matrices, the new bound stake simple a nd applicable form.