seniorLinear Algebra
What makes a matrix ill-conditioned?
Updated May 16, 2026
Short answer
A matrix is ill-conditioned when small input changes cause large output changes.
Deep explanation
Ill-conditioning occurs when eigenvalues span many orders of magnitude or are close to zero. This makes inversion and solving linear systems unstable, leading to unreliable numerical results in ML models and optimization.
Real-world example
Occurs in deep learning when features are highly correlated.
Common mistakes
- Assuming all matrices are equally stable numerically.
Follow-up questions
- How to fix ill-conditioned systems?