Definition:Vandermonde Determinant
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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
- 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