Definition of ill-conditioned linear system

For an invertible square system, the condition number $\kappa(A)=\|A\|\|A^{-1}\|$ describes worst-case relative sensitivity in a chosen induced norm. In the 2-norm, $\kappa_2(A)=\sigma_{\max}/\sigma_{\min}$. A large value means some small relative data perturbations can produce large relative solution changes. Whether a value is problematic depends on required accuracy and input precision, not one universal cutoff.

For $x+y=2$, $(1+\varepsilon)x+y=2+\varepsilon$, the solution is $(1,1)$. Changing the second right side by $\delta$ changes $x$ by $\delta/\varepsilon$.

Conditioning belongs to the problem; stability belongs to the algorithm. A stable solver can still lose accuracy on an ill-conditioned problem. See [condition number](https://knowen.md/nodes/23750).Copy