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Interactive 3D/Condition Number
Matrix A = [[a,b],[c,d]]
a₁₁ = 2.00
a₁₂ = 1.00
a₂₁ = 1.00
a₂₂ = 1.50
Vector b
b₁ = 3.00
b₂ = 2.00
Perturbation δb
δb₁ = 0.10
δb₂ = 0.10
Diagnostics
κ(A):4.13
det(A):2.0000
x* = (1.25, 0.50)
Condition number κ(A) = ‖A‖·‖A⁻¹‖. High κ means nearly parallel lines - a tiny change in b sends x flying. In ML: ill-conditioned Gram matrices make gradient descent unstable.

Condition Number - Interactive Visualization

The condition number κ(A) = ||A|| · ||A⁻¹|| measures how much a linear system amplifies input errors. A high condition number (ill-conditioned matrix) means small errors in b cause large errors in the solution x. Nearly parallel lines in a 2D system give high condition numbers - the intersection point is very sensitive to tiny changes. This is why feature normalization and preconditioning matter in ML.

  • See two line equations that represent Ax=b geometrically
  • Adjust matrix entries and watch condition number κ update
  • Add a small perturbation δb and see the large change in solution δx
  • Watch lines turn red as they become nearly parallel (ill-conditioned)
  • Foundation for understanding why batch normalization helps training stability

Part of the EngineersOfAI Interactive 3D - free interactive visualizations covering every major concept in machine learning and AI engineering. Hover any element for a plain-English explanation. No code required.