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κ

Matrix Condition Number

The condition number κ(A) = ||A||·||A⁻¹|| measures sensitivity to input errors. κ ≈ 1: well-conditioned; κ ≫ 1: ill-conditioned. Hilbert and Vandermonde matrices are famously ill-conditioned.

Concept Fundamentals
κ(A) = ||A||·||A⁻¹||
Formula
κ ≈ 1
Well
κ ≫ 1
Ill
κ(I) = 1
Identity

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Hilbert matrices: κ grows exponentially. κ ≈ 1 for orthogonal matrices. Singular: κ = ∞ (A⁻¹ undefined).

Key quantities
κ(A) = ||A||·||A⁻¹||
Formula
Key relation
κ ≈ 1
Well
Key relation
κ ≫ 1
Ill
Key relation
κ(I) = 1
Identity
Key relation

Ready to run the numbers?

Why: Condition numbers predict numerical stability in linear systems. Ill-conditioned matrices amplify rounding errors.

How: Compute ||A||_F and ||A⁻¹||_F (Frobenius norm). κ = ||A||·||A⁻¹||. Spectral (2-norm) κ uses singular values.

Hilbert matrices: κ grows exponentially.κ ≈ 1 for orthogonal matrices.
Sources:Numerical Linear AlgebraWolfram MathWorld

Run the calculator when you are ready.

Compute Condition NumberUses Frobenius norm; singular matrices are undefined

Examples

Matrix is singular (not invertible)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

κ(AB) ≤ κ(A)·κ(B)

⚡

Frobenius: ||A||_F = √(Σ aᵢⱼ²)

Key Takeaways

  • • κ(A) = ||A||·||A⁻¹|| measures sensitivity of Ax=b to errors.
  • • κ ≈ 1: well-conditioned. κ >> 1: ill-conditioned.
  • • Lose up to log₁₀(κ) digits of accuracy in solution.
  • • κ(I) = 1. Singular matrix: κ = ∞.
  • • Hilbert and Vandermonde matrices are classically ill-conditioned.

Did You Know?

📐Turing introduced condition number in 1948.Source: History
🔢κ(A) ≥ 1 always. κ = 1 only for identity (up to scaling).Source: Bounds
📊Hilbert matrix Hₙ has κ growing exponentially with n.Source: Hilbert
⚛️κ₂(A) = σ_max/σ_min (ratio of singular values).Source: SVD
📜Preconditioning: solve M⁻¹Ax = M⁻¹b to reduce κ.Source: Preconditioning
🔬Ridge regression adds λI to stabilize ill-conditioned XᵀX.Source: ML

How It Works

1. Compute inverse

A⁻¹ exists iff det(A) ≠ 0. Use adjugate/determinant.

2. Compute norms

||A||_F and ||A⁻¹||_F (Frobenius norm).

3. Multiply

κ(A) = ||A||·||A⁻¹||.

4. Interpret

κ < 5: very good. κ > 1000: caution.

Expert Tips

Check before solving

Compute κ before solving Ax=b; high κ = unstable.

Scaling

Scale rows/columns to improve conditioning.

Regularization

Add λI: (A + λI)x = b for stability.

Higher precision

Use double/quad for ill-conditioned systems.

Comparison Table

κ rangeAssessment
κ < 5Very well-conditioned
5 ≤ κ < 100Well-conditioned
100 ≤ κ < 1000Mildly ill-conditioned
κ ≥ 1000Ill-conditioned

FAQ

What is condition number?

κ(A) = ||A||·||A⁻¹||. Measures sensitivity of Ax=b to errors.

When is κ = ∞?

When A is singular (det = 0). No inverse.

What is κ for identity?

κ(I) = 1. Perfect conditioning.

What norm to use?

Frobenius is easy. Spectral (2-norm) is most common in theory.

Hilbert matrix?

Classically ill-conditioned. κ grows exponentially.

How to improve?

Scaling, preconditioning, regularization (A + λI).

Rectangular matrices?

Use pseudoinverse: κ = σ_max/σ_min.

Digits lost?

Up to log₁₀(κ) digits of accuracy lost.

Stats

κ(I)
1
κ(singular)
∞
Digits lost
≤ log₁₀(κ)
Hilbert
κ → ∞

Sources

  • • Wilkinson, The Algebraic Eigenvalue Problem
  • • Golub & Van Loan, Matrix Computations
  • • Trefethen & Bau, Numerical Linear Algebra
  • • Higham, Accuracy and Stability
  • • Turing (1948), Rounding-off Errors
  • • Demmel, Applied Numerical Linear Algebra
Disclaimer: This calculator uses Frobenius norm. For spectral condition number (2-norm), use NumPy. Singular matrices will show an error.
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