Ill-Conditioned & Condition Number

Conditional Statements | Converse, Inverse & Contrapositive

As mentioned in the text, inverse iteration depends on the solution of a system of equations that may be exceedingly ill-conditioned, with condition number on the order of ϵmachine ^-1. We know that it is impossible in general to solve ill-conditioned systems accurately. Is this not a fatal flaw in the algorithm? Show as follows that the answer is no-that ill-conditioning is not a problem in inverse iteration. Suppose A is a real symmetric matrix with one eigenvalue much smaller than the others