Condition Number Calculator

About the Condition Number Calculator

The condition number κ(A) measures how sensitive a linear system Ax = b is to small changes in A or b. A matrix with a low condition number (near 1) is well-conditioned — small perturbations in the input produce proportionally small changes in the output. A large condition number means the system is ill-conditioned and numerical solutions may be unreliable. This calculator computes κ(A) = ‖A‖ · ‖A⁻¹‖ using three different matrix norms: the 1-norm (maximum absolute column sum), the infinity-norm (maximum absolute row sum), and the Frobenius norm (root of the sum of all squared entries). Enter a square matrix up to 5×5, choose from classic presets like the notoriously ill-conditioned Hilbert matrix, and see the condition number, determinant, both the original and inverse matrices, and a color-coded sensitivity gauge. Engineers use condition numbers to assess whether finite-precision arithmetic will produce trustworthy results, and numerical analysts use them to choose appropriate algorithms and preconditioners for large linear systems.

When This Page Helps

Computing a condition number requires finding the matrix inverse and computing norms in multiple formulations — a tedious process for anything larger than 2×2. This calculator handles matrices up to 5×5, computes κ(A) in all three standard norms simultaneously, displays the inverse matrix, and provides a color-coded conditioning gauge. Classic ill-conditioned examples like the Hilbert matrix are available as presets, making it ideal for numerical analysis courses and engineering sensitivity checks.

How to Use the Inputs

1. Set the matrix size using the Size field (1–5 for a square matrix).
2. Enter the matrix values as a comma-separated list, row by row.
3. Select the norm type (1-norm, ∞-norm, or Frobenius) from the dropdown.
4. Click a preset like "Ill-conditioned" or "Hilbert 3×3" to load classic examples.
5. Review the condition number, determinant, matrix norms, and conditioning assessment.
6. Toggle Show Inverse to see the inverse matrix and its norms.
7. Compare condition numbers across all three norms in the summary.

Example Calculation

Tips & Best Practices

• Check that all inputs use the same scale and assumptions before trusting the result.
• Compare the answer with the worked example or a rough estimate to catch entry mistakes.

What the Condition Number Means

The condition number κ(A) = ‖A‖·‖A⁻¹‖ bounds the worst-case amplification of relative errors when solving Ax = b. If κ = 1000, a 0.1% perturbation in the input could cause up to a 100% change in the solution. A well-conditioned matrix (κ close to 1) produces stable solutions; an ill-conditioned matrix (κ ≫ 1) means numerical solutions may lose many digits of accuracy. The identity matrix has κ = 1 in every norm — the theoretical best.

Famous Ill-Conditioned Matrices

The Hilbert matrix H with entries h_ij = 1/(i+j−1) is the classic example of severe ill-conditioning. For a 3×3 Hilbert matrix, κ ≈ 748; for 5×5, κ ≈ 476,000; for 10×10, κ exceeds 10¹³. This exponential growth means that solving Hx = b for large Hilbert matrices is essentially impossible in standard floating-point arithmetic. Vandermonde matrices with closely spaced nodes and matrices arising from polynomial interpolation are also notoriously ill-conditioned.

Norm Choices and Their Interpretations

The 1-norm (maximum absolute column sum) measures the maximum stretch along coordinate axes. The ∞-norm (maximum absolute row sum) is its dual. The Frobenius norm (√Σa²_ij) is an "average" measure that treats all entries equally. For the 2-norm (spectral norm), κ equals the ratio of largest to smallest singular values (σ_max/σ_min). All these norms give condition numbers within a constant factor of each other, so any norm reveals the same qualitative conditioning status.

Frequently Asked Questions

• What does the condition number tell you?

  It bounds the worst-case amplification of relative errors. κ(A) = 1000 means a small perturbation in inputs can be amplified up to 1000× in the output.

• What is a well-conditioned matrix?

  A matrix with κ close to 1. The identity matrix has κ = 1 in any norm — the theoretical best.

• Why are Hilbert matrices ill-conditioned?

  Their entries (1/(i+j−1)) decrease smoothly, making rows nearly linearly dependent. κ grows exponentially with matrix size.

• Does the norm choice matter?

  Different norms give different condition numbers, but they are all within a constant factor of each other for a given matrix size. Any norm reveals the same qualitative conditioning.

• What if the determinant is small but non-zero?

  A small determinant alone does not imply ill-conditioning. The condition number is the proper measure — it accounts for the matrix norm.

• How does condition number relate to singular values?

  The 2-norm condition number equals σ_max / σ_min, the ratio of the largest to smallest singular value.