Square of Vandermonde Matrix

Theorem
The square of the Vandermonde matrix of order $n$:


 * $\mathbf V = \begin{bmatrix}

x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\ x_1^n & x_2^n & \cdots & x_n^n \end{bmatrix}$

is symmetrical in $x_1, \ldots, x_n$.