Value of Vandermonde Determinant

Theorem

Formulation $1$

Let $V_n$ be the Vandermonde determinant of order $n$ defined as the following formulation:

$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}$

Its value is given by:

$\ds V_n = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$

Formulation $2$

Let $V_n$ be the Vandermonde determinant of order $n$ defined as the following formulation:

$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}$

Its value is given by:

$\ds V_n = \prod_{1 \mathop \le j \mathop \le n} x_j \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i}$

Source of Name

This entry was named for Alexandre-Théophile Vandermonde.