Hilbert Matrix

In linear algebra, a Hilbert matrix, introduced by Hilbert (1894), is a square matrix with entries being the unit fractions

For example, this is the 5 × 5 Hilbert matrix:

H = \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 can be regarded as derived from the integral

that is, as a Gramian matrix for powers of x. It arises in the least squares approximation of arbitrary functions by polynomials.

The Hilbert matrices are canonical examples of ill-conditioned matrices, making them notoriously difficult to use in numerical computation. For example, the 2-norm condition number of the matrix above is about 4.8 · 105.

Read more about Hilbert Matrix:  Historical Note, Properties

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