A characteristic of **A** for ill-posed problems is that it has a very large
condition number. In other words, the ill-conditioned matrix **A** is very
near to being singular. When one inverts a matrix which has a very large
condition number, it is highly susceptible to errors. Briefly, the
condition number is defined as or the
ratio of maximum to minimum singular values measured in the norm.
The ideal problem conditioning occurs for orthogonal matrices which have
, while an ill-conditioned matrix will have . The condition of a matrix is relative. It is related
to the precision level of computations and is a function of the size of the
problem. For example, if the condition number exceeds a linear growth rate
with respect to the size of the problem, the problem will become
increasingly ill-conditioned. See [41] for more about the
condition number of matrices.