Vandermonde matrix
A Vandermonde matrix^{} is any $(n+1)\times (n+1)$ matrix of the form
$$\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill {x}_{0}\hfill & \hfill {x}_{0}^{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {x}_{0}^{n}\hfill \\ \hfill 1\hfill & \hfill {x}_{1}\hfill & \hfill {x}_{1}^{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {x}_{1}^{n}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill 1\hfill & \hfill {x}_{n}\hfill & \hfill {x}_{n}^{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {x}_{n}^{n}\hfill \end{array}\right]$$ 
Vandermonde matrices usually arise when considering systems of polynomials evaluated at specific points (i.e. in interpolation^{} or approximation). This may happen, for example, when trying to solve for constants from initial conditions^{} in systems of differential equations or recurrence relations.
Vandermonde matrices also appear in the computation of FFTs (Fast Fourier Transforms). Here the fact that Vandermonde systems $Vz=b$ can be solved in $\mathcal{O}(n\mathrm{log}n)$ flops by taking advantage of their comes into play.
0.1 References

1.
Golub and Van Loan, Matrix Computations, Johns Hopkins University Press 1993
Title  Vandermonde matrix 

Canonical name  VandermondeMatrix 
Date of creation  20130322 13:04:19 
Last modified on  20130322 13:04:19 
Owner  akrowne (2) 
Last modified by  akrowne (2) 
Numerical id  4 
Author  akrowne (2) 
Entry type  Definition 
Classification  msc 65T50 
Classification  msc 65F99 
Classification  msc 15A57 