Vandermonde matrix

A Vandermonde matrix is any $(n+1)\times(n+1)$ matrix of the form

\begin{bmatrix}1&x_{0}&x_{0}^{2}&\cdots&x_{0}^{n}\\ 1&x_{1}&x_{1}^{2}&\cdots&x_{1}^{n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&x_{n}&x_{n}^{2}&\cdots&x_{n}^{n}\end{bmatrix}

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 $\mathcal{O}(n\log n)$ 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	2013-03-22 13:04:19
Last modified on	2013-03-22 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