How bad are Hankel matrices?

Summary.

Considered are Hankel, Vandermonde, and Krylov basis matrices. It is proved that for any real positive definite Hankel matrix of order $n$, its spectral condition number is bounded from below by$3 \cdot 2^{n-6}$ . Also proved is that the spectral condition number of a Krylov basis matrix is bounded from below by$3^{\frac{1}{2}} \cdot 2^{\frac{n}{2}-3}$ . For $V = V(x_1, \ldots, x_n)$, a Vandermonde matrix with arbitrary but pairwise distinct nodes $x_1, \ldots, x_n$, we show that $\mbox{\rm cond}_2\,V \geq 2^{n-2}/n^{\frac{1}{2}}$; if either $|x_j| \leq 1$ or$|x_j| \geq 1$ for all $j$, then $\mbox{\rm cond}_2\,V \geq 2^{n-2}$.