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?