TABLE OF CONTENTS Our hero is the intrepid, yet sensitive matrix A. Our villain is E, who keeps perturbing A. When A is perturbed he puts on a crumpled hat: = A + E.
- G. W. STEWART and JI-GUANG SUN, Matrix Perturbation Theory (1990)
The expression 'ill-conditioned' is sometimes used merely as a term of abuse applicable to matrices or equations ... It is characteristic of ill-conditioned sets of equations that small percentage errors in the coefficients given may lead to large percentage errors in the solution.
- A. M. TURING, Rounding-Off Errors in Matrix Processes (1948)
In this chapter we are concerned with a linear system Ax = b, where 
. In the context of uncertain data or inexact arithmetic there are three important questions:
How much does x change if we perturb A and b; that is, how sensitive is the solution to perturbations in the data?
How much do we have to perturb the data A and b for an approximate solution y to be the exact solution of the perturbed system-in other words, what is the backward error of y?
What bound should we compute in practice for the forward error of a given approximate solution?
To answer these questions we need both normwise and componentwise perturbation theory.
First, we present some classical normwise perturbation results. We denote by any vector norm and the corresponding subordinate matrix norm. As usual,
