  The tau neutrino is the least well measured particle in the Standard Model.
Most notably, the tau neutrino row of the lepton mixing matrix is quite poorly
constrained when unitarity is not assumed. In this paper, we identify data sets
involving tau neutrinos that improve our understanding of the tau neutrino part
of the mixing matrix, in particular $\nu_\tau$ appearance in atmospheric
neutrinos. We present new results on the elements of the tau row leveraging
existing constraints on the electron and muon rows for the cases of unitarity
violation, with and without kinematically accessible steriles. We also show the
expected sensitivity due to upcoming experiments and demonstrate that the tau
neutrino row precision may be comparable to the muon neutrino row in a careful
combined fit.
