Given a matrix, is the Frobenius norm of that matrix always equal to the 2norm 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).

Elementwise 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 = Elementwise 2norm = Schatten 2norm

Induced 2norm = Schatten \inftynorm. This is also called Spectral norm.
So if by “2norm” you mean elementwise or Schatten norm, then they are identical to Frobenius norm. If you mean induced 2norm, you get spectral 2norm, 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 2norm, which they simply call the 2norm. So in that sense, the answer to your question is that the (induced) matrix 2norm 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