TITLE: What is the difference between the Frobenius norm and the 2-norm of a matrix?
QUESTION [66 upvotes]: 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?

REPLY [23 votes]: 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).