Definition:Vandermonde Determinant

Definition

The Vandermonde determinant of order $n$ is the determinant defined as one of the following two formulations:

Formulation 1

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

Formulation 2

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

Also see

• Value of Vandermonde Determinant

• Definition:Vandermonde Matrix

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

• Results about the Vandermonde determinant can be found here.

Source of Name

This entry was named for Alexandre-Théophile Vandermonde.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Vandermonde determinant