Calculator Input
Use the responsive calculator grid below. Large screens show three columns, medium screens show two, and mobile shows one.
Formula Used
Vector L1 norm: ‖x‖₁ = Σ |xᵢ|
Weighted vector L1 norm: ‖x‖₁,w = Σ wᵢ|xᵢ|
Matrix operator L1 norm: ‖A‖₁ = maxⱼ Σᵢ |aᵢⱼ|
Entrywise absolute sum: Σᵢ Σⱼ |aᵢⱼ|
The calculator evaluates absolute values first, then aggregates them according to the selected vector or matrix interpretation.
How to Use This Calculator
- Select whether you are calculating a vector or matrix L1 norm.
- Enter values separated by commas, spaces, or line breaks.
- For matrices, place each row on a new line.
- For weighted vectors, enter matching weights in the optional box.
- Choose the number of decimal places you want displayed.
- Click the calculate button to show the result above the form.
- Review the breakdown table, notes, and Plotly chart.
- Use the download buttons to export the visible result data.
Example Data Table
| Example | Input Type | Sample Input | Expected Interpretation | Sample Result |
|---|---|---|---|---|
| Example A | Vector | 3, -4, 2, -7, 5 | Sum of absolute values | 21 |
| Example B | Weighted Vector | 3, -4, 2 with weights 1, 2, 0.5 | Weighted sum of absolute values | 12 |
| Example C | Matrix | 1 -2 3 / 4 0 -5 / -1 2 6 | Largest absolute column sum | 14 |
Frequently Asked Questions
1. What does the L1 norm measure?
The L1 norm measures total absolute magnitude. For a vector, it adds absolute component values. For a matrix, the operator version finds the largest absolute column sum.
2. How is L1 norm different from L2 norm?
L1 adds absolute values directly. L2 squares values, sums them, and then takes a square root. L1 is often preferred when sparsity or robust deviation measurement matters.
3. Can I use negative numbers?
Yes. Negative values are fully supported because the calculator converts every entry to its absolute value before summing or forming column sums.
4. What is a weighted vector L1 norm?
A weighted L1 norm multiplies each absolute component by a matching non-negative weight. This lets you emphasize or de-emphasize specific dimensions in the final magnitude.
5. How should I enter a matrix?
Enter each row on a new line. Separate values with spaces or commas. All rows must contain the same number of columns for the matrix calculation to work.
6. Why do matrix operator and entrywise results differ?
The operator L1 norm uses only the largest absolute column sum. The entrywise version adds every absolute matrix entry. They answer different mathematical questions.
7. When is the L1 norm useful in practice?
It is useful in optimization, compressed sensing, sparse modeling, error analysis, regularization, and signal processing whenever absolute deviation or sparsity is important.
8. Do the CSV and PDF downloads include the table?
Yes. Both export options include the summary values and the detailed breakdown table currently shown in the calculated result section.