Manuals/calci/HILBERT

MATRIX("HILBERT",order)

• $order$ 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.,$H_{ij}={\frac {1}{i+j-1}}.*Examplefor5x5Hilbertmatrixis:<math><math>{\begin{bmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}\\{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}\\{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}\\{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}\\{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}&{\frac {1}{9}}\\\end{bmatrix}}$.
• 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.