Vandermonde matrix

In Linear Algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a [math]\displaystyle{ (m + 1) \times (n + 1) }[/math] matrix with the form:

[math]\displaystyle{ V = V(x_0, x_1, \cdots, x_m) = \begin{bmatrix} 1 & x_0 & x_0^2 & \dots & x_0^n\\ 1 & x_1 & x_1^2 & \dots & x_1^n\\ 1 & x_2 & x_2^2 & \dots & x_2^n\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 1 & x_m & x_m^2 & \dots & x_m^n \end{bmatrix} }[/math]

with entries [math]\displaystyle{ V_{i,j} = x_i^j }[/math], the j^th power of the number [math]\displaystyle{ x_i }[/math], for all indices [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] where [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] start at 0. Most authors define the Vandermonde matrix as the transpose of the above matrix.

Applications

Vandermonde matrices are commonly used in introductory Linear Algebra courses to prove least squares solutions.