Sandwich Bounds for Gramian and Hankel Spectra

We derive upper and lower bounds for the eigenvalues of the controllability and observability Gramians, as well as for the Hankel singular values, of continuous-time single-input single-output LTI systems with diagonalizable state matrix. The analysis employs a Cauchy matrix formulation that yields explicit, easily computable bounds in terms of the state matrix eigenvalues. The results provide insight into how the Gramian and Hankel spectra decay with the system size. Illustrative examples on structured systems show how the derived bounds are able to capture distinct spectral-scaling regimes.