Suppose that I have two nxn similarity matrices. These matrices contain similarity information between n items. Although both matrices contain similarities of the same n items they do not contain the same similarity values. This might be because the similarities between the items are calculated using different information.
I want to know how similar these matrices are. One simple thing is to find the frobenius distance between the two matrices. But this might be misleading I think.
Are there better ways? What I want to understand whether the structure contained in the two similarity matrices are similar or not.
Let me clarify what I mean by a similarity matrix. Suppose that we have n items. And suppose that each item i is represented with a vector of numbers. Then each element of the similarity matrix $S(i,j) = cosine(v_i, v_j)$ where $v_i$ and $v_j$ are the $ith$ and $jth$ item vectors and $cosine(v_i, v_j)$ is the cosine of the angle between $v_i$ and $v_j$. (Distance metrics other then cosine may also be used)
May be I should have used distance matrix instead of similarity matrix.