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Understanding the correlation structure associated with multiple brain measurements informs about potential "functional groupings" and network organization. The correlation structure can be conveniently captured in a matrix format that summarizes the relationships among a set of brain measurements involving two regions, for example. Such functional connectivity matrix is an important component of many types of investigation focusing on network-level properties of the brain, including clustering brain states, characterizing dynamic functional states, performing participant identification (so-called "fingerprinting") understanding how tasks reconfigure brain networks, and inter-subject correlation analysis. In these investigations, some notion of proximity or similarity of functional connectivity matrices is employed, such as their Euclidean distance or Pearson correlation (by correlating the matrix entries). Here, we propose the use of a geodesic distance metric that reflects the underlying non-Euclidean geometry of functional correlation matrices. The approach is evaluated in the context of participant identification (fingerprinting): given a participant's functional connectivity matrix based on resting-state or task data, how effectively can the participant be identified? Using geodesic distance, identification accuracy was over 95% on resting-state data, and exceeded the Pearson correlation approach by 20%. For whole-cortex regions, accuracy improved on a range of tasks by between 2% and as much as 20%. We also investigated identification using pairs of subnetworks (say, dorsal attention plus default mode), and particular combinations improved accuracy over whole-cortex participant identification by over 10%. The geodesic distance also outperformed Pearson correlation when the former employed a fourth of the data as the latter. Finally, we suggest that low-dimensional distance visualizations based on the geodesic approach help uncover the geometry of task functional connectivity in relation to that during resting-state. We propose that the use of the geodesic distance is an effective way to compare the correlation structure of the brain across a broad range of studies.