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 |