Enter Matrix and Options
Example Data Table
This example uses a 3 × 3 matrix and shows common induced norm results for quick reference.
| Example Matrix | Operator 1-Norm | Operator 2-Norm | Operator Infinity-Norm | Frobenius Norm |
|---|---|---|---|---|
|
[2, -1, 0] [1, 3, 4] [0, -2, 5] |
9.0000 | 6.4497 | 8.0000 | 7.7460 |
Formula Used
Induced operator norm: ||A||p = maxx ≠ 0 ||Ax||p / ||x||p
For the 1-norm: ||A||1 = maxj Σi |aij|, which is the largest absolute column sum.
For the 2-norm: ||A||2 = σmax(A) = √λmax(ATA), which equals the largest singular value.
For the infinity-norm: ||A||∞ = maxi Σj |aij|, which is the largest absolute row sum.
For a custom p value: the page estimates the maximum ratio using generalized power iteration on normalized trial vectors.
How to Use This Calculator
- Choose the matrix dimensions first.
- Paste or type each row on its own line.
- Select 1, 2, infinity, or custom p-norm estimation.
- Set decimal places and advanced iteration controls if needed.
- Press the calculate button to show results above the form.
- Review the metric cards, supporting table, and Plotly graph.
- Download the summary as CSV or PDF when finished.
FAQs
1. What does the operator norm measure?
The operator norm measures the largest stretching effect of a matrix under a chosen vector norm. It tells you how much the matrix can amplify an input vector in the worst case.
2. Why are 1, 2, and infinity norms common?
These norms have practical formulas and strong interpretation. The 1-norm uses column sums, the infinity-norm uses row sums, and the 2-norm equals the largest singular value.
3. Is the 2-norm always the largest norm?
No. Different induced norms measure growth differently. A matrix can have a larger 1-norm or infinity-norm than its 2-norm, depending on how entries are distributed.
4. Why does the custom p option say estimated?
For general p values, an exact closed-form formula is uncommon. The calculator uses an iterative numerical method, so the result is a high-quality estimate rather than a symbolic exact value.
5. Can I use rectangular matrices?
Yes. Operator norms apply to rectangular matrices as well as square ones. The page supports both, provided your matrix text matches the selected row and column counts.
6. What does the singular value chart show?
The chart compares singular values with row and column absolute sums. This helps you see how the selected operator norm relates to spectral behavior and simple sum-based bounds.
7. When should I increase random starts or iterations?
Increase them for custom p values, especially on harder matrices. More starts reduce the chance of missing a strong direction, and more iterations improve numerical convergence.
8. What is the Frobenius norm included for?
The Frobenius norm is a useful comparison metric based on the square root of the sum of squared entries. It helps judge overall matrix magnitude beside induced operator norms.