# Vandermonde Matrix Identity for Hilbert Matrix/Examples

## Examples of Use of Vandermonde Matrix Identity for Hilbert Matrix

### $3 \times 3$ Matrix

Define polynomial root sets $\set {1, 2, 3}$ and $\set {0, -1, -2}$ for Definition:Cauchy Matrix because Hilbert Matrix is Cauchy Matrix.

Illustrate $3\times 3$ case for Vandermonde Matrix Identity for Hilbert Matrix and value of Hilbert matrix determinant:

 $\ds H$ $=$ $\ds {\begin{pmatrix} \frac 1 1 & \frac 1 2 & \frac 1 3 \\ \frac 1 2 & \frac 1 3 & \frac 1 4 \\ \frac 1 3 & \frac 1 4 & \frac 1 5 \\ \end{pmatrix} }$ Hilbert matrix of order $3$

Then:

 $\ds H$ $=$ $\ds -P V_x^{-1} V_y Q^{-1}$ Vandermonde Matrix Identity for Hilbert Matrix $\ds \map \det H$ $=$ $\ds \dfrac 1 {2140}$ Determinant of Matrix Product