> So how does one calculate the condition number of a matrix? Given that real > world systems are large, can we precisely calculate K(A)? Or, can we only > hope to obtain a upper bound for K(A)?
Condition numbers are nice for theoretical considerations, but I haven't seen they have quite a sensible meaning in practical situations. You can calculate them _after_ you have already done a lot of work (i.e. solving the equations) while it is suggested that you should rather know about Condition Numbers _before_ doing so.
With Least Squares problems, though, theory leads to the insight that it is wise to use normed equations, that is: weighted with the trace of the element matrices OR length of the vector of equation coefficients, which is the same thing. This doesn't help in actually predicting how large or how small the C.N. will be, in the end.