Definition:Vandermonde Matrix

Definition
Vandermonde's matrix of order $n$ is a square matrix specified variously as:

etc.
 * $a_{ij} = x_i^{j-1}$
 * $a_{ij} = x_j^i$
 * $a_{ij} = x_i^{n-j}$

Written out in full, it is of the form:


 * $\begin{bmatrix}

1 & x_1 & x_1^2 & \cdots & x_1^{n-2} & x_1^{n-1} \\ 1 & x_2 & x_2^2 & \cdots & x_2^{n-2} & x_2^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^{n-2} & x_n^{n-1} \end{bmatrix}$

or:


 * $\begin{bmatrix}

x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\ x_1^n & x_2^n & \cdots & x_n^n \end{bmatrix}$

or:


 * $\begin{bmatrix}

x_1^{n-1} & x_1^{n-2} & \cdots & x_1 & 1 \\ x_2^{n-1} & x_2^{n-2} & \cdots & x_2 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ x_n^{n-1} & x_n^{n-2} & \cdots & x_n & 1 \\ \end{bmatrix}$

etc.