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# Definition of ill-conditioned linear system

Put content here**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.
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Ill-conditioning is a property of the coefficient matrix $A$, measured by its **condition number**:
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$$\kappa(A) = \|A\| \cdot \|A^{-1}\|$$
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- 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
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**Geometric intuition:** An ill-conditioned system has nearly parallel hyperplanes — a small perturbation in the equations dramatically shifts the intersection point.
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**Example:**
$$\begin{cases} x + y = 2 \\ 1.0001x + y = 2.0001 \end{cases}$$
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The lines are nearly parallel. A tiny change in coefficients produces a large change in the solution.
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**Practical impact:** Ill-conditioned systems are numerically unstable; solutions computed with floating-point arithmetic may be inaccurate.

# Parents

* Linear systems of equations