  Convergence analysis of consensus algorithms is revisited in the light of the
Hilbert distance. Tsitsiklis Lyapunov function is shown to be the Hilbert
distance to consensus in log coordinates. Birkhoff theorem, which proves
contraction of the Hilbert metric for any positive homogeneous monotone map,
provides an early yet general convergence result for consensus algorithms.
Because Birkhoff theorem holds in arbitrary cones, we extend consensus
algorithms to the cone of positive definite matrices. The proposed
generalization finds applications in the convergence analysis of quantum
stochastic maps, which are a generalization of stochastic maps to
non-commutative probability spaces.
