Documentation

# norm

Norm of matrix or vector

## Syntax

``norm(A)``
``norm(A,p)``
``norm(V)``
``norm(V,P)``

## Description

example

````norm(A)` returns the `2`-norm of matrix `A`. Because symbolic variables are assumed to be complex by default, the norm can contain unresolved calls to `conj` and `abs`.```

example

````norm(A,p)` returns the `p`-norm of matrix `A`.```
````norm(V)` returns the `2`-norm of vector `V`.```

example

````norm(V,P)` returns the `P`-norm of vector `V`.```

## Examples

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Compute the `2`-norm of the inverse of the 3-by-3 magic square `A`:

```A = inv(sym(magic(3))) norm2 = norm(A)```
```A = [ 53/360, -13/90, 23/360] [ -11/180, 1/45, 19/180] [ -7/360, 17/90, -37/360] norm2 = 3^(1/2)/6```

Use `vpa` to approximate the result with 20-digit accuracy:

`vpa(norm2, 20)`
```ans = 0.28867513459481288225```

Compute the norm of `[x y]` and simplify the result. Because symbolic variables are assumed to be complex by default, the calls to `abs` do not simplify.

```syms x y simplify(norm([x y]))```
```ans = (abs(x)^2 + abs(y)^2)^(1/2)```

Assume `x` and `y` are real, and repeat the calculation. Now, the result is simplified.

```assume([x y],'real') simplify(norm([x y]))```
```ans = (x^2 + y^2)^(1/2)```

Remove assumptions on `x` for further calculations. For details, see Use Assumptions on Symbolic Variables.

`assume(x,'clear')`

Compute the `1`-norm, Frobenius norm, and infinity norm of the inverse of the 3-by-3 magic square `A`:

```A = inv(sym(magic(3))) norm1 = norm(A, 1) normf = norm(A, 'fro') normi = norm(A, inf)```
```A = [ 53/360, -13/90, 23/360] [ -11/180, 1/45, 19/180] [ -7/360, 17/90, -37/360] norm1 = 16/45 normf = 391^(1/2)/60 normi = 16/45```

Use `vpa` to approximate these results to 20-digit accuracy:

```vpa(norm1, 20) vpa(normf, 20) vpa(normi, 20)```
```ans = 0.35555555555555555556 ans = 0.32956199888808647519 ans = 0.35555555555555555556```

Compute the `1`-norm, `2`-norm, and `3`-norm of the column vector ```V = [Vx; Vy; Vz]```:

```syms Vx Vy Vz V = [Vx; Vy; Vz]; norm1 = norm(V, 1) norm2 = norm(V) norm3 = norm(V, 3)```
```norm1 = abs(Vx) + abs(Vy) + abs(Vz) norm2 = (abs(Vx)^2 + abs(Vy)^2 + abs(Vz)^2)^(1/2) norm3 = (abs(Vx)^3 + abs(Vy)^3 + abs(Vz)^3)^(1/3)```

Compute the infinity norm, negative infinity norm, and Frobenius norm of `V`:

```normi = norm(V, inf) normni = norm(V, -inf) normf = norm(V, 'fro')```
```normi = max(abs(Vx), abs(Vy), abs(Vz)) normni = min(abs(Vx), abs(Vy), abs(Vz)) normf = (abs(Vx)^2 + abs(Vy)^2 + abs(Vz)^2)^(1/2)```

## Input Arguments

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Input, specified as a symbolic matrix.

One of these values `1`, `2`, `inf`, or `'fro'`.

• `norm(A,1)` returns the `1`-norm of `A`.

• `norm(A,2)` or `norm(A)` returns the `2`-norm of `A`.

• `norm(A,inf)` returns the infinity norm of `A`.

• `norm(A,'fro')` returns the Frobenius norm of `A`.

Input, specified as a symbolic vector.

• `norm(V,P)` is computed as `sum(abs(V).^P)^(1/P)` for `1<=P<inf`.

• `norm(V)` computes the `2`-norm of `V`.

• `norm(A,inf)` is computed as `max(abs(V))`.

• `norm(A,-inf)` is computed as `min(abs(V))`.

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### 1-Norm of a Matrix

The `1`-norm of an m-by-n matrix A is defined as follows:

### 2-Norm of a Matrix

The `2`-norm of an m-by-n matrix A is defined as follows:

The `2`-norm is also called the spectral norm of a matrix.

### Frobenius Norm of a Matrix

The Frobenius norm of an m-by-n matrix A is defined as follows:

`${‖A‖}_{F}=\sqrt{\sum _{i=1}^{m}\left(\sum _{j=1}^{n}{|{A}_{ij}|}^{2}\right)}$`

### Infinity Norm of a Matrix

The infinity norm of an m-by-n matrix A is defined as follows:

`${‖A‖}_{\infty }=\mathrm{max}\left(\sum _{j=1}^{n}|{A}_{1j}|,\text{\hspace{0.17em}}\sum _{j=1}^{n}|{A}_{2j}|,\dots ,\sum _{j=1}^{n}|{A}_{mj}|\right)$`

### P-Norm of a Vector

The `P`-norm of a 1-by-n or n-by-1 vector V is defined as follows:

`${‖V‖}_{P}={\left(\sum _{i=1}^{n}{|{V}_{i}|}^{P}\right)}^{1}{P}}$`

Here n must be an integer greater than 1.

### Frobenius Norm of a Vector

The Frobenius norm of a 1-by-n or n-by-1 vector V is defined as follows:

`${‖V‖}_{F}=\sqrt{\sum _{i=1}^{n}{|{V}_{i}|}^{2}}$`

The Frobenius norm of a vector coincides with its `2`-norm.

### Infinity and Negative Infinity Norm of a Vector

The infinity norm of a 1-by-n or n-by-1 vector V is defined as follows:

The negative infinity norm of a 1-by-n or n-by-1 vector V is defined as follows:

## Tips

• Calling `norm` for a numeric matrix that is not a symbolic object invokes the MATLAB® `norm` function.