# Matrix Norm Calculator

URL: https://speedycalculator.com/calculators/math/matrix-norm/
Last updated: 2026-10-30

> The matrix norm calculator computes the 1-norm, ∞-norm, 2-norm (spectral norm), Frobenius norm and max norm of a 2 × 2, 2 × 3, 3 × 2 or 3 × 3 matrix.

## Inputs

- Matrix size (rows × columns)
- a₁
- b₁
- c₁
- a₂
- b₂
- c₂
- a₃
- b₃
- c₃

## Results

- 2-norm (spectral)
- 1-norm
- ∞-norm
- Frobenius norm
- Max norm

## Formula

```
‖A‖₁ = max column sum of |aᵢⱼ|,  ‖A‖∞ = max row sum,  ‖A‖₂ = largest singular value
‖A‖_F = √(Σ aᵢⱼ²),  ‖A‖max = max |aᵢⱼ|
```

## Explanation

Matrix norms measure the "size" of a matrix. They bound errors in numerical algorithms, define condition numbers and measure distances between matrices in data science.

### How to use the Matrix Norm Calculator

- Choose the **matrix size** – the matching grid of fields appears.
- Enter the entries **row by row** (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The column and row sums give the 1- and ∞-norms directly. The 2-norm is the largest singular value, found from the eigenvalues of AᵀA. The Frobenius norm is the Euclidean length of all entries.

## Worked example

Inputs — Matrix size (rows × columns): 2 × 2; a₁: 1; b₁: -2; a₂: 3; b₂: 4.

Results — **2-norm (spectral) = 5.116672736**; 1-norm = 6; ∞-norm = 7; Frobenius norm = 5.477225575; Max norm = 4.

## Key terms

- **Matrix norm**: A measure of the size of a matrix.

## FAQ

**Which matrix norm should I use?**
The 2-norm for theory, the 1- or ∞-norm for quick bounds, Frobenius for least squares.

**Is the max norm submultiplicative?**
No – ‖AB‖ ≤ ‖A‖‖B‖ can fail for it.

**How are the norms related?**
‖A‖₂ ≤ ‖A‖_F ≤ √r ‖A‖₂ for rank r.

**What is an induced norm?**
A matrix norm defined from a vector norm, such as the 1-, 2- and ∞-norms.


## Sources

- [Wolfram MathWorld – Matrix norm](https://mathworld.wolfram.com/MatrixNorm.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)