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:

- 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 $V$?
- Are there known lower and upper bounds which relate the singular values of $V$ and $\tilde{V}$ to one another?