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^® software 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

norm - Vector and matrix norms

Syntax

Description

Remarks

See Also