How to compare two similarity matrices? Hi,
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.
Thanks
Ahmet
 A: EDIT: I naively thought similarity matrix == dissimilarity matrix, this isn't the case. It's been too long since I did bioinformatics. My answer below should properly say "dissimilarity matrix satisfying the triangle inequality". Such a matrix can be constructed along the lines in the comments above.
A similarity matrix is just a metric on a finite space. The standard metric on the space of all finite metric spaces is the Gromov-Hausdorff metric.
A: As far as you use the cosine as similarity measure, the matrix is a correlation matrix. For this situation in statistics there is the concept of "canonical correlation", and this might be then the most appropriate for your case: it gives an index how much "variance of one set of variables is explained by the other". The two set of variables are the two sets of vectors $\small v_i $ here.     
Another option could be to compute the cholesky factors ("factor loadings matrices") L1 and L2 of each of the correlation matrices R1 and R2 and do a target-rotation of L1 to L2. Then, for instance, the squared distances of the vector-tips of each related vector in rotated(L1) and L2 could be summed and this could be understood as similarity measure of the matrices(!) - but this is no standard method as far as I know... 
