Distance/Similarity between two matrices

Distance/Similarity between two matrices

I'm in the process of writing an application which identifies the closest
matrix from a set of square matrices $M$ to a given square matrix $A$. The
closest can be defined as the most similar.
I think finding the distance between two given matrices is a fair approach
since the smallest Euclidean distance is used to identify the closeness of
vectors.
I found that the distance between two matrices ($A,B$) could be calculated
using the Frobenius distance $F$:
$$F_{A,B} = \sqrt{trace((A-B)*(A-B)')} $$
where $B'$ represents the conjugate transpose of B.
I have the following points I need to clarify
Is the distance between matrices a fair measure of similarity?
If distance is used, is Frobenius distance a fair measure for this
problem? any other suggestions?