norm (MATLAB Functions)

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))).

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)).

`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

See Also

cond, condest, normest, rcond, svd

nonzeros normest