Matrix Condition Number Calculator

Enter Matrix Data

Matrix entries

Example format: 4 2 0
2 5 1
0 1 3

Summary norm Displayed decimals Power iterations

Numerical tolerance

Quick examples

This page accepts square matrices up to 8 × 8. Larger systems are possible, but server-side elimination becomes heavier.

Example Data Table

Example matrix	Determinant	κ₁(A)	κ_∞(A)	κ_F(A)	Approx. κ₂(A)
`[4 2 0; 2 5 1; 0 1 3]`	44.000000	4.000000	4.000000	4.684245	3.366025
This sample is invertible and reasonably well-conditioned, so small data perturbations usually stay manageable.

Formula Used

Condition number definition: κ(A) = ||A|| · ||A^-1||

1-norm: ||A||₁ = max column sum of absolute values

Infinity-norm: ||A||_∞ = max row sum of absolute values

Frobenius norm: ||A||_F = √(Σ a_ij²)

Approximate 2-norm: ||A||₂ ≈ √(largest eigenvalue of A^TA)

A larger condition number means the matrix is more sensitive to small input or rounding changes. Singular matrices have infinite condition numbers because no inverse exists.

This calculator computes the inverse by Gauss–Jordan elimination with partial pivoting, then multiplies matrix and inverse norms to estimate sensitivity.

How to Use This Calculator

Enter a square matrix in the textarea, one row per line.
Separate entries with spaces, commas, or semicolons.
Choose which norm should act as the summary result.
Set the displayed decimals, iteration count, and tolerance.
Press Calculate Condition Number to show results above the form.
Review norms, inverse, assessment, and the graph.
Use the CSV or PDF buttons to export the calculated report.

FAQs

1. What does a matrix condition number show?

It measures how strongly output values can react to tiny input changes. Small condition numbers indicate stable systems. Large values warn that rounding, measurement noise, or perturbations may cause noticeable solution errors.

2. Why does a singular matrix return infinity?

A singular matrix has no inverse. Since the condition number formula multiplies the matrix norm by the inverse norm, the result becomes infinite or numerically undefined.

3. Which norm should I use?

Use the 2-norm for a common sensitivity summary, the 1-norm for column-based bounds, the infinity-norm for row-based bounds, and the Frobenius norm for overall energy-style magnitude.

4. What is a good condition number?

Values near 1 are excellent. Values under 10 are usually comfortable. Values above 100 suggest sensitivity, while very large values often signal ill-conditioning and unreliable numerical solutions.

5. Why estimate digits lost?

A rough rule uses log₁₀(κ). If κ is 1,000, about three digits of accuracy may be vulnerable. It is an estimate, not a guaranteed error count.

6. Does scaling rows or columns change conditioning?

Yes. Rescaling can improve or worsen conditioning depending on the matrix. Practical numerical workflows often scale systems before solving to reduce sensitivity and improve stability.

7. Why use power iteration for the 2-norm?

The 2-norm equals the largest singular value. Power iteration gives a practical estimate by repeatedly applying A^TA, which is efficient for small web calculators.

8. Can I analyze non-square matrices here?

This page is designed for square matrices because inverse-based condition numbers need an invertible square matrix. Rectangular matrices require singular-value or pseudoinverse-based approaches.