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

hilbert1

In Scilab, you can generate such a matrix by typing in the following commands:

--> Hinv5=testmatrix('hilb',5);H5=inv(Hinv5)

Hilbert (1894) introduced this matrix to study the following question in approximation theory:

“Assume that I = [a, b], is a real interval. Is it then possible to find a non-zero polynomial P with integral coefficients, such that the following integral is smaller than any given bound ε > 0, taken arbitrarily small?”

hilbert3

Read more on Wikipedia.