Condition number: κ(A) = ||A||·||A⁻¹||. For 2-norm: κ₂ = σ_max/σ_min. κ ≈ 1: well-conditioned (stable). κ ≫ 1: ill-conditioned (small errors amplify). In 16-digit arithmetic, Ax=b loses ~log₁₀(κ) digits of accuracy.
How to Use
Enter a matrix:
- A: Square matrix
- Output: κ₁, κ₂, κ∞
- Analysis: Digit loss estimate
Interpretation
κ = 1: perfect (orthogonal matrices). κ = 10³: lose ~3 digits. κ = 10¹⁵: result is meaningless in double precision. Hilbert matrices are famously ill-conditioned: κ grows exponentially with size.
Different Norms
- κ₁: max column sum norm
- κ₂: σ_max/σ_min (most useful)
- κ∞: max row sum norm
- All related: κ₁/n ≤ κ₂ ≤ n·κ₁
Step-by-Step Instructions
- 1Enter a matrix.
- 2View κ in all norms.
- 3See singular values.
- 4Check digit loss.
- 5Assess stability.