Manuals/calci/HILBERT

MATRIX("HILBERT",order)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle order} is the order of the Hilbert matrix.

Description

This function gives matrix of order 3x3 with the property of Hilbert.
A Hilbert matrix, is a square matrix with entries being the unit fractions. i.e.,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{ij}=\frac{1}{i+j-1}. *Example for 5x5 Hilbert matrix is: <math>\begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\ \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \\ \end{bmatrix}} .
The Hilbert matrix is an example of a Hankel matrix.
The Hilbert matrix is symmetric and positive definite.
Also Hilbert matrices are canonical examples of ill-conditioned matrices, making them notoriously difficult to use in numerical computation.
Here MATRIX("hilbert") gives the hilbert matrices with a decimal places .
i.e., For 1/2 it will show 0.5, 1/3 will show 0.333 and so on.