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?
Best Answer
There are three important types of matrix norms. For some matrix $A$
Induced norm, which measures what is the maximum of $\frac{\|Ax\|}{\|x\|}$ 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.