Definition:Vandermonde Matrix
(Redirected from Definition:Vandermonde's Matrix)
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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
- Definition:Alternant Matrix and Definition:Alternant Determinant, of which the Vandermonde matrix and Vandermonde determinant are examples
- Results about Vandermonde matrices can be found here.
Source of Name
This entry was named for Alexandre-Théophile Vandermonde.