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
  

See Also

cond, condest, normest, rcond, svd

nonzeros

normest