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2 Direct and Inverse Problem Formulation     continued...

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.

(See exercises 8 and 8.)