What is the difference between the Frobenius norm and the 2-norm of a matrix?

Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results?

If they are identical, then I suppose the only difference between them is the method of calculation, eh?

Answer

There are three important types of matrix norms. For some matrix A

  • Induced norm, which measures what is the maximum of for any x \neq 0 (or, equivalently, the maximum of \|Ax\| for \|x\|=1).

  • Element-wise norm, which is like unwrapping A into a long vector, then calculating its vector norm.

  • Schatten norm, which measures the vector norm of the singular values of A.

So, to answer your question:

  • Frobenius norm = Element-wise 2-norm = Schatten 2-norm

  • Induced 2-norm = Schatten \infty-norm. This is also called Spectral norm.

So if by “2-norm” you mean element-wise or Schatten norm, then they are identical to Frobenius norm. If you mean induced 2-norm, you get spectral 2-norm, which is \le Frobenius norm. (It should be less than or equal to)

As far as I can tell, if you don’t clarify which type you’re talking about, induced norm is usually implied. For example, in matlab, norm(A,2) gives you induced 2-norm, which they simply call the 2-norm. So in that sense, the answer to your question is that the (induced) matrix 2-norm is \le than Frobenius norm, and the two are only equal when all of the matrix’s eigenvalues have equal magnitude.

Attribution
Source : Link , Question Author : Ricket , Answer Author : Brandon J. DeHart

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