Sum of Elements in Inverse of Vandermonde Matrix

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}$

Let $V_n^{-1}$ be its inverse, from Inverse of Vandermonde Matrix:
 * $b_{i j} = \begin{cases}

\paren {-1}^{j - 1} \paren {\dfrac{\displaystyle \sum_{\substack {1 \mathop \le m_1 \mathop < \ldots \mathop < m_{n - j} \mathop \le n \\ m_1, \ldots, m_{n - j} \mathop \ne i} } x_{m_1} \cdots x_{m_{n - j} } } {x_i \displaystyle \prod_{\substack {1 \mathop \le m \mathop \le n \\ m \mathop \ne i} } \paren {x_m - x_i} } } & : 1 \le j < n \\ \qquad \qquad \qquad \dfrac 1 {x_i \displaystyle \prod_{\substack {1 \mathop \le m \mathop \le n \\ m \mathop \ne i} } \paren {x_i - x_m} } & : j = n \end{cases}$

The sum of all the elements of $V_n^{-1}$ is:
 * $\displaystyle \sum_{1 \mathop \le i, \ j \mathop \le n} b_{i j} = 1 - \prod_{k \mathop = 1}^n \paren {1 - \dfrac 1 {x_k} }$

Proof
Sum of Elements of Invertible Matrix implies the sum of elements in $V_n^{-1}$ is:


 * $ 1 - \det \paren { V_n^{-1} } \det \paren { V_n - J_n } $.

The plan is to expand $\det \paren { V_n - J_n } $ and simplify.

The method is efficiently communicated in the $3\times 3$ case:

The proof is completed by induction using the lemmas below.

Lemma 1
Let:


 * $\displaystyle A_n = \det \paren { \begin{smallmatrix}

1 & \cdots & 1 \\ x_1 & \cdots & x_n \\ \vdots & \cdots & \vdots \\ x_1^{n-1} & \cdots & x_n^{n-1} \\ \end{smallmatrix} }$

Then:


 * $\displaystyle \det \paren { V_n } = \paren { \prod_{k \mathop = 1}^n x_k } \det \paren { A_n }$  Effect of Elementary Row Operations on Determinant

Lemma 2

 * $\displaystyle \det(V_n - J_n) = \prod_{k \mathop = 1}^n \paren { x_k - 1 }

\det \paren { A_n }$

Lemma 3

 * $\displaystyle \det \paren { V_n^{-1} } \det(V_n - J_n) =

\dfrac { \prod_{k \mathop = 1}^n \paren { x_k - 1 }  } { \prod_{k \mathop = 1}^n { x_k } }$  by Lemmas 1 and 2.