I would like to test that two difference/distance/dissimilarity matrices are not the same. i.e. the rows and columns between the two matrices represent the same features, but the distances are obtained from 2 populations and I'm interested in whether the difference matrices "look different" between the populations.
I'm think I'm looking for something similar to a Mantel's test, but with the null hypothesis flipped. Whereas (as I understand it) the Mantel test tests for a linear correlation between two dissimilarity matrices against a null hypothesis of no linear correlation, the null in my case is that the two dissimilarity matrices are the same, and I'm interested in rejecting that null when two dissimilarity matrices differ significantly.
As a follow up to this question, once I have some sort of omnibus test for difference, what would be the best way to decompose the differences to contributions from individual cells.