📊 Condition Number Calculator
Calculate the condition number κ(A) = ||A|| × ||A⁻¹||
How to Use This Calculator
Enter Matrix
Input all elements of your 2×2 square matrix.
Calculate
Click to compute κ(A) = ||A|| × ||A⁻¹||.
Interpret Results
Small κ ≈ 1: well-conditioned. Large κ: ill-conditioned (near-singular).
Formula
κ(A) = ||A|| × ||A⁻¹||
Where ||·|| is a matrix norm (Frobenius, 2-norm, etc.)
Definition:
Condition number measures how sensitive the solution of Ax = b is to changes in A or b.
Bounds:
κ(A) ≥ 1 (always)
κ(A) = 1 if A is orthogonal/unitary
κ(A) = ∞ if A is singular
Interpretation:
- κ < 10: Well-conditioned (stable)
- 10 ≤ κ < 100: Moderately conditioned
- 100 ≤ κ < 1000: Ill-conditioned
- κ ≥ 1000: Very ill-conditioned (near-singular)
Error Amplification:
If input has relative error ε, output can have error up to κ × ε
About Condition Number Calculator
The Condition Number Calculator computes κ(A) = ||A|| × ||A⁻¹||, a measure of numerical stability. A large condition number indicates that small changes in input can cause large changes in output, making the matrix "ill-conditioned" and prone to numerical errors.
When to Use This Calculator
- Numerical Analysis: Assess stability of matrix operations
- Linear Systems: Predict error in solving Ax = b
- Inverse Problems: Evaluate sensitivity to perturbations
- Optimization: Check conditioning of Hessian matrices
- Machine Learning: Assess numerical stability
Why Use Our Calculator?
- ✅ Complete Analysis: Shows norm, inverse norm, and condition number
- ✅ Classification: Categorizes matrix as well/ill-conditioned
- ✅ Inverse Display: Shows A⁻¹ if it exists
- ✅ Educational: Helps understand numerical stability
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Well-Conditioned: Small κ, stable solutions
- Ill-Conditioned: Large κ, sensitive to perturbations
- Singular: κ = ∞ (matrix is not invertible)
- Error Bound: ||Δx||/||x|| ≤ κ × (||ΔA||/||A|| + ||Δb||/||b||)
- Best Case: κ = 1 for orthogonal/unitary matrices
Applications
Linear Systems: If κ(A) is large, small errors in A or b cause large errors in solution x.
Least Squares: High condition number indicates unreliable solutions.
Frequently Asked Questions
What is condition number?
Condition number κ(A) = ||A|| × ||A⁻¹|| measures numerical stability. A large condition number means the matrix amplifies errors in calculations.
What does a large condition number mean?
A large condition number (κ > 100) indicates the matrix is "ill-conditioned" - small changes in input cause large changes in output, making it numerically unstable.
Can condition number be less than 1?
No, κ(A) ≥ 1 always. For orthogonal/unitary matrices, κ = 1 (best possible). The condition number equals 1 for identity matrices.
What if condition number is infinite?
If κ(A) = ∞, the matrix is singular (determinant = 0) and not invertible. This is the worst case for numerical stability.
Which norm is used?
This calculator uses the Frobenius norm: ||A|| = √(Σᵢⱼ aᵢⱼ²). Other common choices are the 2-norm (spectral norm) or 1-norm (column sum norm).