Velocity ellipsoid (${\sigma }_{ij}^2=⟨(v_i-{\bar{v}}_i)(v_j-{\bar{v}}_j)⟩$)

The velocity ellipsoid represents the positive velocity covariance tensor by its orthogonal principal axes and eigenvalue square roots. The uncentered second-moment tensor differs from covariance when a population streams. Vanishing mixed second moments align the second-moment ellipsoid with the chosen frame; for pure azimuthal streaming in a phase-mixed axisymmetric model, the off-diagonal covariance also vanishes.