Calculator Input
Example Data Table
These sample cases show typical norm outputs for quick checking.
| Case | Input | Norm | Expected Result |
|---|---|---|---|
| Vector | [3, -4] | L1 | 7 |
| Vector | [3, -4] | L2 | 5 |
| Matrix | [[1, 2], [-3, 4]] | Frobenius | √30 ≈ 5.4772 |
| Matrix | [[1, 2], [-3, 4]] | Infinity | 7 |
Formula Used
Vector Norms
L1 norm: ||x||₁ = Σ |xᵢ|
L2 norm: ||x||₂ = √(Σ xᵢ²)
L∞ norm: ||x||∞ = max |xᵢ|
Lp norm: ||x||ₚ = (Σ |xᵢ|ᵖ)^(1/p), where p ≥ 1.
Matrix Norms
Frobenius norm: ||A||F = √(Σ Σ aᵢⱼ²)
1-norm: ||A||₁ = maxⱼ Σᵢ |aᵢⱼ|
Infinity norm: ||A||∞ = maxᵢ Σⱼ |aᵢⱼ|
Max absolute entry: max |aᵢⱼ|
Spectral 2-norm: ||A||₂ = √(λmax(AᵀA))
Entrywise p-norm: (Σ Σ |aᵢⱼ|ᵖ)^(1/p)
How to Use This Calculator
- Select Vector or Matrix.
- Choose the norm type you need.
- Enter a valid p value when using a p-based norm.
- Paste your values into the input box.
- Set decimal places and spectral power iterations if required.
- Enable normalization when you want the scaled output.
- Click Calculate Norm to display the result above the form.
- Use the CSV and PDF buttons to export the result section.
Frequently Asked Questions
1) What is a norm?
A norm measures the size of a vector or matrix. It converts many values into one nonnegative quantity that helps compare magnitude, stability, and sensitivity across problems.
2) When should I use the L1 norm?
Use L1 when you want the sum of absolute values. It is common in sparse modeling, robust optimization, and situations where every component contributes linearly.
3) Why is the L2 norm popular?
The L2 norm reflects Euclidean length. It is widely used in geometry, least squares, machine learning, and physics because it connects naturally to distance and energy.
4) What does the matrix 1-norm represent?
The matrix 1-norm is the largest absolute column sum. It highlights the strongest column-wise accumulation of magnitude and is useful in analysis and conditioning studies.
5) What does the matrix infinity norm represent?
The infinity norm is the largest absolute row sum. It focuses on row-wise magnitude and is often used in stability checks and iterative method analysis.
6) Is the spectral norm exact here?
The calculator estimates the spectral 2-norm with power iteration. For most practical cases it is very accurate, and more iterations usually improve the estimate.
7) What happens when normalization is enabled?
The calculator divides each vector or matrix entry by the selected norm value. This creates a scaled version whose chosen norm becomes one, when possible.
8) Can I export the results?
Yes. Use the CSV button for table data and the PDF button for a printable snapshot of the result area, including summaries and steps.