Matrix Conditioning

Numerical Analysis · Axiom Academy

Understanding when linear systems are sensitive to errors Even with exact arithmetic, real-world data has measurement errors. A well-conditioned matrix gives stable solutions despite small errors. An ill-conditioned matrix amplifies errors dramatically. The condition number κ(A) measures how much errors in b can be amplified in the solution x: The relative error bound tells us: Geometrically, ill-conditioning means the rows of A are nearly parallel (2D) or nearly coplanar (3D). The intersection point becomes very sensitive to small changes. The condition number depends on which matrix norm we use: What To Do About Ill-Conditioning Strategies for ill-conditioned systems: Scale the matrix: Equilibration can help with extreme row/column scales Use higher precision: Quad precision or iterative refinement Regularization: Add λI to make the problem better-conditioned Different formulation: Normal equations Aᵀ Ax = Aᵀb squares κ; use QR instead You've learned how matrix conditioning affects solution accuracy!

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