Definition:Vandermonde Matrix

From ProofWiki
Jump to navigation Jump to search

Definition

The Vandermonde matrix of order $n$ is a square matrix specified in the following formulations:

Formulation 1

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


Formulation 2

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


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.