Inverse of Vandermonde Matrix/Corollary

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Corollary to Inverse of Vandermonde Matrix

Define for variables $\set {y_1,\ldots, y_k}$ elementary symmetric functions:

$\ds \map {e_m} {\set {y_1, \ldots, y_k} }$

$=$

$\ds \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le k } y_{j_1} y_{j_2} \cdots y_{j_m}$

for $m = 0, 1, \ldots, k$

Let $\set {x_1, \ldots, x_n}$ be a set of distinct values.

Let $W_n$ and $V_n$ be Vandermonde matrices of order $n$:

$W_n = \begin{bmatrix} 1 & x_1 & \cdots & x_1^{n-1} \\ 1 & x_2 & \cdots & x_2^{n-1} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & x_1^{n-1} & \cdots & x_n^{n-1} \\ \end{bmatrix}, \quad V_n = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\ x_1^n & x_2^n & \cdots & x_n^n \\ \end{bmatrix}$

Let their matrix inverses be written as $W_n^{-1} = \begin{bmatrix} b_{ij} \end{bmatrix}$ $V_n^{-1} = \begin{bmatrix} c_{ij} \end{bmatrix}$.

Then:

$\ds b_{ij}$

$=$

$\ds \dfrac {\paren {-1}^{n - i} \map {e_{n - i} } {\set {x_1, \ldots, x_n} \setminus \set {x_j} } } {\prod_{m \mathop = 1, m \mathop \ne j }^n \paren {x_j - x_m} }$

for $i, j = 1, \ldots, n$

$\ds c_{ij}$

$=$

$\ds \dfrac 1 {x_i} \, b_{j i}$

for $i, j = 1, \ldots, n$

Proof

The details appear in Inverse of Vandermonde Matrix/Proof 1, same notation. $\blacksquare$