seniorLinear Algebra
Why is matrix inversion considered numerically unstable in practice?
Updated May 16, 2026
Short answer
Matrix inversion amplifies numerical errors, especially for ill-conditioned matrices.
Deep explanation
Computing A⁻¹ explicitly is unstable because small floating-point errors in A get amplified, especially when A has a high condition number. Instead of inversion, numerical systems solve Ax=b directly using factorization methods like LU or QR decomposition to avoid instability.
Real-world example
Used in ML training where stability is more important than exact inversion.
Common mistakes
- Explicitly computing inverse instead of solving linear systems.
Follow-up questions
- Why does condition number matter?