Birkhoff's contraction coefficient (Q1887619)

\textit{G. Birkhoff} [Trans. Am. Math. Soc. 85, 219--227 (1957; Zbl 0079.13502); Lattice theory. Third (new) ed. (1967; Zbl 0153.02501); pp. 383--386] proved that multiplication by any positive square matrix induces a contraction mapping on positive projective space with respect to the Hilbert projective metric, and then derived the Perron-Frobenius theorem. \textit{D. Hilbert} [Math. Ann. 57, 137--150 (1903; JFM 34.0525.01)] had originally introduced this metric for a different purpose. Birkhoff reduced the problem to the case of a \(2\times 2\) matrix and then computed the contraction coefficient using ideas from projective geometry. \textit{E. Seneta} [Non-negative matrices and Markov chains. 2nd ed. (1981; Zbl 0471.60001); pp. 100--110] presented a second proof which relies on no established theory and is quite long and complicated. \textit{R. Cavazos-Cadena} [Linear Algebra Appl. 375, 291--297 (2003; Zbl 1048.15018)] gave a third proof which is much easier and shorter than the second and uses only elementary calculus. In this paper the author provides a further proof using simple algebra, calculus and linear programming. Unlike the earlier proofs, it verifies, rather than derives, the value of the contraction coefficient, but it demonstrates its minimality constructively and tightens the contraction inequality.