Matrix Norm Calculator
Compute the Frobenius norm, 1-norm, or infinity-norm of a matrix. Choose the norm type from the dropdown.
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Notes
Matrix Norms
A matrix norm measures the 'size' of a matrix. Different norms capture different geometric properties.
| Norm | Formula | Description |
|---|---|---|
| Frobenius | √(Σᵢⱼ aᵢⱼ²) | Square root of sum of squared entries |
| 1-norm | max column: Σᵢ|aᵢⱼ| | Maximum absolute column sum |
| ∞-norm | max row: Σⱼ|aᵢⱼ| | Maximum absolute row sum |
Example: Frobenius Norm
| 1 | 2 |
| 3 | 4 |
‖A‖_F = √(1² + 2² + 3² + 4²) = √(1 + 4 + 9 + 16) = √30 ≈ 5.477
The Frobenius norm is the most commonly used in practice. It equals √tr(A^T A) — the square root of the trace of A^T A.
See also
- Matrix Calculations Guide — Matrix Norm — In-depth notes on Matrix Norm with worked examples
Frequently Asked Questions
What is the Frobenius norm?
The Frobenius norm is ‖A‖_F = √(sum of squares of all entries). It equals √tr(A^T A).
When would you use the infinity-norm?
The ∞-norm gives the maximum absolute row sum, useful when bounding the maximum deviation in linear systems.
Are all matrix norms equivalent?
In finite dimensions, all matrix norms are equivalent in the sense that they bound each other by constants. However, the constants depend on the matrix size.