An alternative derivation of Birkhoff’s formula for the contraction coefficient of a positive matrix

This note concerns the projective contraction coefficient τ(H) of a rectangular matrix H with positive entries. A simple proof of an explicit formula for τ(H), originally established by [Trans. Am. Math. Soc. 85 (1957) 219], is given. The motivation for this work comes from the area of Markov decision processes, and the argument is based on elementary differential calculus.