Value of Vandermonde Determinant/Formulation 2

Theorem

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:

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

Proof

The proof follows directly from that for Value of Vandermonde Determinant/Formulation 1 and the result Determinant with Row Multiplied by Constant.

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Source of Name

This entry was named for Alexandre-Théophile Vandermonde.

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