Definition:Vandermonde Determinant/Formulation 1

Definition

The Vandermonde determinant of order $n$ is the determinant defined as:

$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}$

Also presented as

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

Ones at Right

$V_n = \begin {vmatrix}

{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 {vmatrix}$

Ones at Top

$V_n = \begin {vmatrix}

    1   &    1    & \cdots &  1 \\
  x_1   &  x_2    & \cdots &  x_n \\
{x_1}^2 & {x_2}^2 & \cdots & {x_n}^2 \\
 \vdots &  \vdots & \ddots & \vdots \\
{x_1}^{n - 2} & {x_2}^{n - 2} & \cdots & {x_n}^{n - 2} \\
{x_1}^{n - 1} & {x_2}^{n - 1} & \cdots & {x_n}^{n - 1}

\end {vmatrix}$

Also see

• Value of Vandermonde Determinant

• Results about the Vandermonde determinant can be found here.

Source of Name

This entry was named for Alexandre-Théophile Vandermonde.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Vandermonde determinant