The Many Proofs and Applications of Perron’s Theorem

This paper chronicles the wide dispersal of Perron's 1907 result on positive matrices into many fields of science. The many proofs given during the last 93 years are categorized and critiqued (including Perron's original two proofs), and a more natural proof is presented. This simple-to-understand result of Perron provides a unequaled vehicle for taking students on a tour of many applied areas with some depth.