MATLAB Function Reference
norm

Vector and matrix norms

Syntax

• ```n` `=` `norm(A)
n` `=` `norm(A,p)
```

Description

The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix. The `norm` function calculates several different types of matrix norms:

```n = norm(A) ``` returns the largest singular value of `A`, `max(svd(A))`.

```n = norm(A,p) ``` returns a different kind of norm, depending on the value of `p.`

 If `p` is... Then `norm` returns... `1` The 1-norm, or largest column sum of `A`, `max(sum(abs(A))`. `2` The largest singular value (same as `norm(A)`). `inf` The infinity norm, or largest row sum of `A`, `max(sum(abs(A')))`. `'fro'` The Frobenius-norm of matrix `A`, `sqrt(sum(diag(A'`*`A)))`.

When `A` is a vector:

 `norm(A,p)` Returns `sum(abs(A).^p)^(1/p)`, for any 1 `<=` `p` `<=` . `norm(A)` Returns `norm(A,2)`. `norm(A,inf)` Returns `max(abs(A))`. `norm(A,-inf)` Returns `min(abs(A))`.

Remarks

Note that `norm(x)` is the Euclidean length of a vector `x`. On the other hand, MATLAB uses "length" to denote the number of elements `n` in a vector. This example uses `norm(x)/sqrt(n)` to obtain the root-mean-square (RMS) value of an `n`-element vector `x`.

• ```x = [0 1 2 3]
x =
0     1     2     3

sqrt(0+1+4+9)   % Euclidean length
ans =
3.7417

norm(x)
ans =
3.7417

n = length(x)   % Number of elements
n =
4

rms = 3.7417/2  % rms = norm(x)/sqrt(n)
rms =
1.8708
```

`cond`, `condest`, `normest`, `rcond`, `svd`