norm

Purpose

Vector and matrix norms.

Synopsis

n = norm(X)
n = norm(X,p)
n = norm(X,'fro')

Description

The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix. Several different types of norms can be calculated:

n = norm(X), where X is a matrix, is the largest singular value of X.

n = norm(X,p), lets you specify a value to indicate largest singular value, largest column sum, or largest row sum of matrix X:

  • norm(X,1) is the 1-norm, or largest column sum of X, max(sum(abs((X))).
  • norm(X,2) is the same as norm(X).
  • norm(X,inf) is the infinity norm, or largest row sum of X, max(sum(abs(X'))).
    n = norm(X,'fro') is the F-norm of matrix X, sqrt(sum(diag(X'*X))).

    When the X is a vector, slightly different rules apply:

  • norm(x,p) = sum(abs(x).^p)^(1/p).
  • norm(x) = norm(x,2).
  • norm(x)/sqrt(n) is the root-mean-square (RMS) value.
  • norm(x,inf) = max(abs(x)).
  • norm(x, -inf) = min(abs(x)).

    Examples

    For a matrix

    A =
    1    2    3
    4    5    6
    7    8    9
           
norm(A)       = 16.8481
norm(A,1)     = 18
norm(A,2)     = 16.8481
norm(A,inf)   = 24
norm(A,'fro') = 16.8819
    For a vector

    v =
    1    2    3
          
norm(v)       = 3.7417
norm(v,1)     = 6
norm(v,2)     = 3.7417
norm(v,Inf)   = 3
norm(v,pi)    = 3.2704

    See Also

    cond, max, min, rcond, svd

    (c) Copyright 1994 by The MathWorks, Inc.