# norm

Vector and matrix norms

## Syntax

• `n = norm(X)` example
• `n = norm(X,p)` example
• `n = norm(X,'fro')` example

## Description

example

````n = norm(v)` returns the 2-norm or Euclidean norm of vector `v`.```

example

````n = norm(v,p)` returns the vector norm defined by `sum(abs(v)^p)^(1/p)`, where `p` is any positive real value, `Inf`, or `-Inf`. If `p` is `Inf`, then `n = max(abs(v))`.If `p` is `-Inf`, then `n = min(abs(v))`.```

example

````n = norm(X)` returns the 2-norm or maximum singular value of matrix `X`.```

example

````n = norm(X,p)` returns the p-norm of matrix `X`, where `p` is `1`, `2`, or `Inf`.```

example

````n = norm(X,'fro')` returns the Frobenius norm, `sqrt(sum(diag(X'*X)))`.```

## Examples

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### 1- and 2- Norm of Vector

Calculate the 2-norm of a vector corresponding to the point (-2,3,-1) in 3-D space. The 2-norm is equal to the Euclidean length of the vector.

```X = [-2 3 -1]; n = norm(X)```
```n = 3.7417```

Calculate the 1-norm of the vector, which is the sum of the element magnitudes.

`n = norm(X,1)`
```n = 6```

### 2-Norm of Matrix

Calculate the 2-norm of a matrix, which is the largest singular value.

```X = [2 0 1;-1 1 0;-3 3 0]; n = norm(X)```
```n = 4.7234```

### Frobenius Norm of Sparse Matrix

Use `'fro'` to calculate the Frobenius norm of a sparse matrix, which calculates the 2-norm of the column vector, `S(:)`.

```S = sparse(1:25,1:25,1); n = norm(S,'fro')```
```n = 5```

## Input Arguments

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### `v` — Input vectorvector

Input vector.

Data Types: `single` | `double`
Complex Number Support: Yes

### `X` — Input matrixmatrix

Input matrix. Use `norm(X,'fro')` when `X` is sparse.

Data Types: `single` | `double`
Complex Number Support: Yes

### `p` — Norm type2 (default) | positive integer scalar | `Inf` | `-Inf`

Norm type, specified as `2` (default), a different positive integer scalar, `Inf`, or `-Inf`. The valid values of `p` and what they return depend on whether the first input to `norm` is a matrix or vector, as shown in the table.

 Note:   This table does not reflect the actual algorithms used in calculations.
pMatrixVector
`1``max(sum(abs(X)))``sum(abs(X))`
`2``max(svd(X))``sum(abs(X).^2)^(1/2)`
Positive, real-valued numeric `p``sum(abs(X).^p)^(1/p)`
`Inf``max(sum(abs(X')))``max(abs(X))`
`-Inf``min(abs(X))`

## Output Arguments

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### `n` — Matrix or vector normscalar

Matrix or vector norm, returned as a scalar. The norm gives a measure of the magnitude of the elements. By convention, `norm` returns `NaN` if the input contains `NaN` values.