Definition:Vandermonde Matrix

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


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

etc.

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}

1 & 1  & \cdots  &   1 \\ x_1 & x_2 & \cdots & x_n \\ \vdots & \vdots & \cdots & \vdots \\ x_1^{n - 1} & x_2^{n - 1} & \cdots & 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.

Also known as
A Vandermonde matrix is often seen referred to as Vandermonde's matrix.

The first form is preferred on because it is slightly less grammatically unwieldy than the possessive style.

Also see

 * Definition:Alternant Matrix and Definition:Alternant Determinant, of which the Vandermonde matrix and Vandermonde determinant are examples