User:Hjilderda

Theorem

Let $V_n$ be the Vandermonde matrix of order $n$ given by:

$V_n = \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}$

Then its inverse $V_n^{-1} = \sqbrk b_n$ can be specified as:

$b_{ij} = \dfrac {\ds \sum_{\stackrel {1 \le k_1 < \ldots < k_{n - j} \le n} {k_1, \ldots, k_{n - j} \ne i} } \paren {-1}^{j - 1} x_{k_1} \ldots x_{k_{n - j} } } {\ds x_i \prod_{\stackrel {1 \le k \le n} {k \ne i} } \paren {x_k - x_i} }$