I have multiple 20x20 matrices that represent histogram distributions. I want to compare these matrices with one another, and I'd prefer a metric that would return a scalar metric that correlates to the degree of similarity between two histograms other than a euclidean L"n" distance. Similarity can be proportional or inversely to the magnitude of the scalar, as long as it's consistent. Any ideas? I know similarity is very ambiguous, but I would like somewhere to start.
By "20 X 20 matrices" I assume you mean that you have bi-variate distributions (rows and columns are bin ranges for each of two variables, while cell values are counts for each bin). I also assume that your bin sizes are identical for all matrices.
What you are looking for is a "statistical distance": "A distance between populations can be interpreted as measuring the distance between two probability distributions and hence they are essentially measures of distances between probability measures." https://en.wikipedia.org/wiki/Statistical_distance
The first answer to this question should help you: How to measure similarity of bivariate probability distributions?
Here's another question and answers that has related information: https://stackoverflow.com/questions/10002261/kullback-leibler-divergence-as-histogram-distance-function