Square of Vandermonde Matrix

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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 $x_1, \ldots, x_n$.


Proof