Linear Algebra, 5th Edition

6.10* Conditioning and the Rayleigh Quotient

In Section 3.4, we studied specific techniques that allow us to solve systems of linear equations in the form $Ax=b$, where A is an $m\times n$ matrix and b is an $m\times 1$ vector. Such systems often arise in applications to the real world. The coefficients in the system are frequently obtained from experimental data, and, in many cases, both m and n are so large that a computer must be used in the calculation of the solution. Thus two types of errors must be considered. First, experimental errors arise in the collection of data since no instruments can provide completely accurate measurements. Second, computers introduce roundoff errors. One might intuitively feel that small relative changes in the coefficients of ...