Condition Numbers and the Loss of Precision of Linear Equations

Abstract

The condition number of an invertible real or complex n x n matrix A is defined as

$$ \kappa (A) = \left\| A \right\|\;\left\| {{{A}^{{ - 1}}}} \right\|, $$

where ‖A‖ is the operator norm

$$ \left\| A \right\| = \mathop{{\sup }}\limits_{{x \ne 0}} \frac{{\left\| {Ax} \right\|}}{{\left\| x \right\|}} $$

and ℝⁿ or ℂⁿ is given the usual inner product. The condition number measures the relative error in the solution of the system of linear equations

$$ {\text{Ax = v}}{\text{.}} $$