Definition:Vandermonde Matrix/Formulation 1

Definition

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

$\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}$

That is, such that:

$a_{i j} = {x_i}^{j - 1}$

Also presented as

The Vandermonde matrix of order $n$ can be presented in various orientations, for example:

Ones at Right

$\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}$

Ones at Top

$\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}$

Also known as

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

The first form is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$ because it is slightly less grammatically unwieldy than the possessive style.

Also see

• Definition:Vandermonde Matrix/Formulation 2
• Definition:Vandermonde Determinant

• Results about Vandermonde matrices can be found here.

Source of Name

This entry was named for Alexandre-Théophile Vandermonde.