L2 norm or Frobenius norm?

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Xiaohan Du
Xiaohan Du on 31 Aug 2018
Answered: Christine Tobler on 18 Sep 2018
Hi all,
I read that Matlab norm(x, 2) gives the 2-norm of matrix x, is this the L2 norm of x? Some people say L2 norm is square root of sum of element square of x, but in Matlab norm(x, 2) gives max singular value of x, while norm(x, 'fro') gives square root of sum element square.
If I want to do |x|||_2^2, should I use (norm(x, 2))^2 or (norm(x, 'fro'))^2?
Many thanks!
  1 Comment
Adam on 31 Aug 2018
doc norm
gives the maths of each of the options.

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Answers (2)

Christine Tobler
Christine Tobler on 18 Sep 2018
The L2-norm of a matrix, |A|||_2, ( norm(A, 2) in MATLAB) is an operator norm, which is computed as max(svd(A)).
For a vector x, the norm |x|||_2, ( norm(x, 2) in MATLAB), is a vector norm, defined as sqrt(sum(x.^2)).
The Frobenius norm |A|||_F, ( norm(A, 'fro') in MATLAB), is equivalent to a vector norm applied to all elements of the matrix A. This is identical to norm(A(:), 2).
See the Wikipedia page on matrix norms for more information.
By the way, if the matrix A is of size 1-by-n or n-by-1, the matrix norm and vector norm interpretations give the same result (max(svd(x)) is identical to sqrt(sum(x.^2))).

Yuvaraj Venkataswamy
Yuvaraj Venkataswamy on 31 Aug 2018
Use 'fro' to estimate the Frobenius norm of a matrix, which estimates the 2-norm of the matrix.
if true
n = norm(x,'fro');


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