  This paper introduces $K$-Tensors, a novel self-consistent clustering
algorithm designed to cluster positive semi-definite (PSD) matrices by their
eigenstructures. Clustering PSD matrices is crucial across various fields,
including computer and biomedical sciences. Traditional clustering methods,
which often involve matrix vectorization, tend to overlook the inherent PSD
characteristics, thereby discarding valuable shape and eigenstructural
information. To preserve this essential shape and eigenstructral information,
our approach incorporates a unique distance metric that respects the PSD nature
of the data. We demonstrate that $K$-Tensors is not only self-consistent but
also reliably converges to a local optimum. Through numerical studies, we
further validate the algorithm's effectiveness and explore its properties in
detail.
