Scaled Almost Diagonal Matrices with Multiple Singular Values

Abstract

New estimate for relative distances between the singular values and the moduli of the appropriate diagonal elements of a scaled almost diagonal square matrix are derived. In case of a multiple singular value the bounds 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 gaps between singular values. The new estimates refine the existing ones which are based on the absolute gaps between singular values. They are especially appropriate for the smallest singular values. For triangular and essentially triangular matrices, the new bounds take simple and applicable form.