TITLE: Distance/Similarity between two matrices
QUESTION [61 upvotes]: 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?

REPLY [7 votes]: If we have two matrices $A,B$.
Distance between $A$ and $B$ can be calculated using Singular values or $2$ norms.
You may use Distance $= \vert(\text{fnorm}(A)-\text{fnorm}(B))\vert$
where fnorm = sq root of sum of squares of all singular values.