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?

Solution 1:

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.

Solution 2:

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).