Definition:Vandermonde Matrix

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


Also see

  • Results about Vandermonde matrices can be found here.


Source of Name

This entry was named for Alexandre-Théophile Vandermonde.


Sources