This paper presents a method of symmetric positive semidefinite (SPSD) matrices classification and its application to the analysis of structural brain networks (connectomes). Structural connectomes are modeled as weighted graphs in which edge weights are proportional to the number of streamline connections between brain regions detected by a tractography algorithm. The construction of structural brain networks does not typically guarantee that their adjacency matrices lie in some topological space with known properties. This makes them differ from functional connectomes-correlation matrices representing co-activation of brain regions, which are usually symmetric positive definite (SPD). Here, we propose to transform structural connectomes by taking their normalized Laplacians prior to any analysis, to put them into a space of symmetric positive semidefinite (SPSD) matrices, and apply methods developed for manifold-valued data. The geometry of the SPD matrix manifold is well known and used in many classification algorithms. Here, we expand existing SPD matrix-based algorithms to the SPSD geometry and develop classification pipelines on SPSD normalized Laplacians of structural connectomes. We demonstrate the performance of the proposed pipeline on structural brain networks reconstructed from the Alzheimer‘s Disease Neuroimaging Initiative (ADNI) data.