An alternative derivation of Birkhoff's formula for the contraction coefficient of a positive matrix. (Q1414717)

Author' s abstract: This note concerns the projective contraction coefficient \({\tau}(H)\) of a rectangular matrix \(H\) with positive entries. A simple proof of an explicit formula for \({\tau}(H)\), originally established by \textit{G. Birkhoff} [Trans. Am. Math. Soc. 85, 219--227 (1957; Zbl 0079.13502)], is given. The motivation for this work comes from the area of Markov decision processes, and the argument is based on elementary differential calculus.