It is common in biological applications to work with data presented as correlation matrices, and it is often the case that the observed variables are distorted by an unknown monotonic nonlinearity. In such settings, the usual linear algebra-based tools for analysis of matrices can fail to detect structure of interest, or even give false impressions of its existence.

By analogy to the Jordan canonical form, we introduce the "order canonical form" of a symmetric matrix, which retains all of the invariant information in the matrix under an element-wise action of the group of monotonic increasing functions. Using the tools of persistent homology, we extract from the order complex a family of signatures of geometric structure (or lack thereof) in the elements of the matrix. As an application, we study the pairwise correlations in the neuronal population of the hippocampus in rats under a variety of behavioral conditions and find strong evidence of a geometric structure in the functional connectivity of the network.