History

1907

  • Perron writes Habilitationsschrift

  • Markov presents on chains

The topic of Perron's paper was the convergence criteria of partial sums which corresponded to the coefficients of Jacobi’s algorithm. Perron first introduced a 2×2 matrix to the problem and considered the characteristic roots. To continue towards his conclusions on convergence, Perron used limits of ratios of the elements in the matrix to find properties of the eigenvalue ρ. It followed that, in some cases, an eigenvalue ρ0 must be the largest absolute value root or have the largest multiplicity. Then, he found that the necessary case relates to strict nonnegative requirements. After considering this problem, Perron wrote a lemma and a corollary that was very close to what we call Perron’s theorem. He then published a new paper, Towards the Theory of Matrices, which included Perron’s theorem and his proof (Hawkins).

Perron's proof was noted to be longer than ideal and brought in ideas that were not strictly linear algebra.

1908

  • Frobenius proves Perron's theorem

  • Markov publishes paper on chains

Frobenius was an expert on matrix theory, so he likely came to know of Perron’s theorem when Perron published his paper "Towards the Theory of Matrices" (Hawkins). Frobenius published a proof for Perron’s theorem that avoided the dreaded limit lemma in 1908. He also considered what could be said if the matrix was nonnegative, instead of strictly positive.

1909

  • Frobenius publishes second related paper

In this paper, published in 1909, Frobenius gave an even stronger proof of the theorem which used inner product notation. With this technique, he also proved other propositions of a similar vein (Hawkins).

1912

  • Frobenius publishes paper answering 1909 paper

  • Markov's paper is translated into German

This year, Frobenius published another paper which started with the exploration of the possible characteristic roots for a nonnegative matrix where nonnegative eigenvectors exist. This led to the separation between primitive and non-primitive irreducible matrices in his theorem conditions. Frobenius included one application in his paper. However, Perron-Frobenius theory was not yet seen as clearly applicable (Hawkins).

1931

  • Perron-Frobenius Theory is connected with Markov chains

Markov chains are one of the earliest applications of Perron-Frobenius theory. Even so, it took until the 1930s for interest in Markov chains to become widespread. Consequently, Markov chains were developed thoroughly and in more generality, including non-negative stochastic matrices. However, Markov chains and Perron-Frobenius theory still were not linked. Richard von Mises made a close connection in 1931 in his book The Calculus of Probabilities and its Application to Statistics and Theoretical Physics (Hawkins). He used Frobenius’ conclusions in his application and was aware of the connection to Markov Chains.

1936

  • Perron-Frobenius Theory is explicitly linked with Markov chains

It was V.I. Romanovsky who cited Frobenius in relation to Markov chains in his “Investigations on Markoff chains” (Hawkins).

1945

  • Leslie publishes model

Patrick Leslie introduced a model for growth of a stratified population, a population that can be divided into subgroups. We are interested only in the case where all the si>0 and bn>0 (Sternberg, Meyer). The Leslie Model uses Perron-Frobenius theory in order to find the steady state for the stratified population.

1950

  • Collatz-Wielandt formula added to Perron-Frobenius theorem

"In 1942 the German mathematician Lothar Collatz (1910–1990) discovered the following formula for the Perron root, and in 1950 Helmut Wielandt (p. 534) used it to develop the Perron–Frobenius theory." (Meyer)

Many other additions have been made since then.

1998

  • Google creators develop PageRank

Sergey Brin and Larry Page developed Google’s PageRank. It is very well know now, even acknowledging that it was not the first page ranking algorithm. The algorithm is designed to place value on a webpage based on the links between pages (Emerging Technology from the arXiv). This algorithm gets very complicated, quickly. Perron-Frobenius theory lets us eliminate some time consuming matrix power calculations.

Theorem progression

Table 1 Comparison of early versions of Perron's theorem