Value of Vandermonde Determinant
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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.