A set of matrices often used---and occasionally misused---as examples in matrix calculations is the set of Hilbert matrices. One situation in which they occur is the following:
Suppose a continuous function 
is given on the interval 
and we are asked to approximate 
by a polynomial of degree n - 1
in x. We write the polynomial in the form
and define the error in the approximation to be
The coefficients 
are determined by the requirement that E be
minimized. Since the error is a differentiable function of the unknowns 
,
at the minimum
Evaluating these derivatives leads to the conditions
Interchanging the summation and integration, we obtain
There are n equations to be satisfied by the n unknowns 
. If we let
and
then the equations can be written as:
Thus the column of coefficients 
can
be found by solving the 
system
where the matrix 
has elements
and the vector 
is determined by the
given function 
.
The matrix 
is the 
Hilbert matrix. We will let
denote its inverse,
We are primarily interested in Hilbert matrices because they are very badly conditioned, even for small values of n, and because their condition number grows rapidly with n. Some of the values are shown in the table below:
The ill-conditioning nature of the Hilbert matrices can be traced back to the
approximation problem which we used to introduce them. On the interval
the functions 
are
very nearly linearly dependent. This means that the rows of the Hilbert
matrix are very linearly dependent, i.e., that the matrix is very nearly
singular. In such cases, a small perturbations in the data can result in
large perturbations in the answers. In the original problem, small
errors in the function 
or rounding errors in its calculation
can result in large changes in the coefficients 
. In short, the
approximation problem is not ``well-posed" when it is in a form that
leads to a matrix like the Hilbert matrix.