  Kolmogorov $n$-widths and Hankel singular values are two commonly used
concepts in model reduction. Here we show that for the special case of linear
time-invariant dynamical (LTI) systems, these two concepts are directly
connected. More specifically, the greedy search applied to the Hankel operator
of an LTI system resembles the minimizing subspace for the Kolmogorov n-width
and the Kolmogorov $n$-width of an LTI system equals its $(n+1)st$ Hankel
singular value once the subspaces are appropriately defined. We also establish
a lower bound for the Kolmorogov $n$-width for parametric LTI systems and
illustrate that the method of active subspaces can be viewed as the dual
concept to the minimizing subspace for the Kolmogorov $n$-width.

Systems and Control;Numerical Analysis