Calculator Inputs
Formula Used
General definition: κ(A) = ||A|| × ||A-1||
1-norm: κ₁(A) = ||A||₁ × ||A-1||₁
Infinity norm: κ∞(A) = ||A||∞ × ||A-1||∞
Frobenius norm: κF(A) = ||A||F × ||A-1||F
Approximate 2-norm: κ₂(A) ≈ σmax(A) / σmin(A)
Here, σmax is the largest singular value and σmin is the smallest singular value. Larger condition numbers indicate greater sensitivity to data or rounding changes.
How to Use This Calculator
- Select the matrix size that matches your square matrix.
- Paste matrix values into the input box.
- Choose the norm you want to prioritize.
- Adjust decimals, iterations, and tolerance if needed.
- Press the calculate button to view the results.
- Review the inverse, determinant, norms, and conditioning label.
- Use the chart to compare condition numbers across norms.
- Export the results as CSV or PDF when needed.
Example Data Table
| Matrix | Determinant | κ₁(A) | κ∞(A) | κF(A) | κ₂(A) approx |
|---|---|---|---|---|---|
| [[4, 1], [2, 3]] | 10 | 3 | 3 | 3 | 2.618034 |
This sample is stable and well-conditioned. It is useful for testing the form, export buttons, and graph output quickly.
FAQs
1. What does a matrix condition number measure?
It measures how sensitive a linear system or matrix calculation is to small input changes. A larger value means errors can grow faster.
2. Why must the matrix be invertible?
The standard condition number uses A-1. If the matrix has no inverse, the condition number is undefined or effectively infinite.
3. Which norm should I choose?
Use the 1-norm for column-based analysis, the infinity norm for row-based analysis, the Frobenius norm for overall energy, and the 2-norm for singular value sensitivity.
4. What does a large condition number mean?
A large value suggests numerical instability. Small rounding errors or small data changes may create large output differences during solving or inversion.
5. Why is the 2-norm result approximate?
This file estimates extreme singular values numerically. That gives a strong practical result without requiring external linear algebra libraries.
6. Can I enter decimal and negative values?
Yes. The parser accepts integers, decimals, and negative numbers. Separate entries with commas or spaces, and place each row on a new line.
7. Does scaling a matrix affect conditioning?
Uniform scalar multiplication does not change the exact condition number. Uneven scaling between rows or columns can change it significantly.
8. Is the determinant enough to judge stability?
No. A nonzero determinant only confirms invertibility. A matrix can still be invertible yet highly ill-conditioned and sensitive to small errors.