Definition of ill-conditioned linear system

Definition: A linear system $Ax = b$ is ill-conditioned if small changes in the input (coefficients or right-hand side) produce large changes in the solution.

Ill-conditioning is a property of the coefficient matrix $A$, measured by its condition number:

$$\kappa(A) = \|A\| \cdot \|A^{-1}\|$$

• If $\kappa(A)$ is large (e.g., $\gt 10^6$), the system is ill-conditioned
• If $\kappa(A)$ is close to 1, the system is well-conditioned

Geometric intuition: An ill-conditioned system has nearly parallel hyperplanes — a small perturbation in the equations dramatically shifts the intersection point.

Example:
$$\begin{cases} x + y = 2 \\ 1.0001x + y = 2.0001 \end{cases}$$

The lines are nearly parallel. A tiny change in coefficients produces a large change in the solution.

Practical impact: Ill-conditioned systems are numerically unstable; solutions computed with floating-point arithmetic may be inaccurate.