Vandermonde Matrix Identity for Hilbert Matrix/Examples

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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