Let $V(x\_1,..,x\_n)$ be the Vandermonde matrix induced by $x\_1,..,x\_n$, and 
let $\tilde{V} := V(\frac{x\_1}{h},...,\frac{x\_n}{h})$.
My intuition says that the condition number should be invariant under such scaling of the nodes at least for some special cases of node configurations. My question is then: 


1. Are there constraints on the nodes 
$x\_1,..,x\_n$ (i.e symmetric, positive, equally spaced or others)
for which ${\cal K}(\tilde{V})= {\cal K}(V)$, where
${\cal K}(V)$ is the condition number for a matrix $V$?
2. Are there known lower and upper bounds which relate the singular 
values of $V$ and $\tilde{V}$ to one another?



The condition number of a scaled Vandermonde matrix
For any configuration of nodes, as $h\to \infty$, $\tilde{V}$ tends to a singular matrix while remaining bounded in norm, so $\mathcal K(\tilde{V})\to \infty$. 


