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?
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$\begingroup$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.
$\endgroup$ $\begingroup$The 2-norm (spectral norm) of a matrix is the greatest distortion of the unit circle/sphere/hyper-sphere. It corresponds to the largest singular value (or |eigenvalue| if the matrix is symmetric/hermitian).
The Forbenius norm is the "diagonal" between all the singular values.
i.e. $$||A||_2 = s_1 \;\;,\;\;||A||_F = \sqrt{s_1^2 +s_2^2 + ... + s_r^2}$$
(r being the rank of A).
Here's a 2D version of it: $x$ is any vector on the unit circle. $Ax$ is the deformation of all those vectors. The length of the red line is the 2-norm (biggest singular value). And the length of the green line is the Forbenius norm (diagonal).
See Wikipedia for all definitions. Take this matrix:$$ \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} $$Its Frobenius norm is $\sqrt{10}$, but its eigenvalues are $3,1$ so, if the matrix is symmetric, its $2$-norm is the spectral radius, i.e., $3$. The Frobenius norm is always at least as large as the spectral radius. The Frobenius norm is at most $\sqrt{r}$ as much as the spectral radius, and this is probably tight (see the section on equivalence of norms in Wikipedia).
Note that the Schatten $2$-norm is equal to the Frobenius norm.
$\endgroup$ 4 $\begingroup$The L2 (or L^2) norm is the Euclidian norm of a vector.
The Frobenius norm is the Euclidian norm of a matrix.
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