Inverse of Vandermonde Matrix/Proof 2

Definition 1

 * $V_n

= \begin{bmatrix} x_1      & \cdots & x_n \\ x_1^2    & \cdots & x_n^2 \\ \vdots   & \ddots & \vdots \\ x_1^{n}  & \cdots & x_n^{n} \\ \end{bmatrix} ,\quad V = \begin{bmatrix} 1        & \cdots & 1 \\ x_1      & \cdots & x_n \\ \vdots   & \ddots & \vdots \\ x_1^{n-1} & \cdots & x_n^{n-1} \\ \end{bmatrix} \quad $ Vandermonde matrices


 * $D =

\begin{bmatrix} x_1   & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0     & \cdots & x_n \\ \end{bmatrix}, \quad P = \begin{bmatrix} \map {p_1} {x_1} & \cdots & 0 \\ \vdots          & \ddots & \vdots \\ 0               & \cdots & \map {p_n} {x_n} \\ \end{bmatrix} \quad $ Definition:Diagonal Matrix


 * $ E =

\begin{bmatrix} E_{11} & \cdots & E_{1n} \\ \vdots & \ddots & \vdots \\ E_{n1} & \cdots & E_{nn} \\ \end{bmatrix} \quad $ Matrix of symmetric functions


 * where for $\mathbf {1 \mathop \le i,j \mathop \le n}$:

Lemma 2

 * $ EV = P$


 * $ V^{-1} = P^{-1} E\quad$ provided $\set {x_1,\ldots,x_n}$ is a set of distinct values.


 * $ V_n^{-1} = D^{-1} V^{-1}\quad$ provided $\set {x_1,\ldots,x_n}$ is a set of distinct values, all nonzero.

Proof of Lemma 2

Matrix multiply establishes $EV=P$, provided:


 * $(1)\quad \sum_{k = 1}^n E_{ik}\, x_j^{k-1} = \begin{cases} 0 & i \neq j \\ \map {p_i} {x_i} & i = j \end{cases}

$ Polynomial $\map {p_i} {x}$ is zero for $x \in \set {x_1,\ldots,x_n} \setminus \set {x_i}$. Then (1) is equivalent to


 * $(2)\quad \sum_{k = 1}^n

\paren { -1 }^{n-k} e_{n-k} \paren {    \set {x_1,\ldots,x_n} \setminus \set {x_i} } \, x_j^{k-1} = \map {p_i} {x_j} $

Apply Viete's Formulas to degree $n-1$ monic polynomial $\map {p_i} {\mathbf {u} }$:


 * $(3)\quad \sum_{k = 1}^{n} \paren {-1}^{n-k}

e_{n-k} \paren {    \set {x_1,\ldots,x_n} \setminus \set {x_i} } \, {\mathbf {u} }^{k-1} = \map {p_i} {\mathbf {u} } $

Substitute ${\mathbf {u} } = x_j$ into (3), proving (2) holds.

Then (2) and (1) hold, proving $EV=P$.

Assume $\set {x_1,\ldots,x_n}$ is a set of distinct values.

Then $\det \paren {P}$ is nonzero.

The Invertible Matrix Theorem implies $P$ has an inverse $P^{-1}$.

Multiply $EV=P$ by $P^{-1}$, then:


 * $V^{-1} = P^{-1} E\quad$ Left or Right Inverse of Matrix is Inverse

Similarly, $D^{-1}$ exists provided $\set {x_1,\ldots,x_n}$ is a set of nonzero values.

Then $V_n^{-1} = D^{-1} V^{-1}$ by Lemma 1.

Proof of the Theorem

Assume $\set {x_1,\ldots,x_n}$ is a set of distinct values.

Let $d_{ij}$ denote the entries in $V^{-1}$. Then:

Then:

Assume $\set {x_1,\ldots,x_n}$ is a set of distinct values, all nonzero.

Let $b_{ij}$ denote the entries in $V_n^{-1}$. Then:

Factor $\paren {-1}^{n-1}$ from the denominator of (4) to agree with Knuth (1997).