Continued fractions and the origins of the Perron–Frobenius theorem |
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Authors: | Thomas Hawkins |
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Institution: | (1) Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA |
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Abstract: | The theory of nonnegative matrices is an example of a theory motivated in its origins and development by purely mathematical
concerns that later proved to have a remarkably broad spectrum of applications to such diverse fields as probability theory,
numerical analysis, economics, dynamical programming, and demography. At the heart of the theory is what is usually known
as the Perron–Frobenius Theorem. It was inspired by a theorem of Oskar Perron on positive matrices, usually called Perron’s
Theorem. This paper is primarily concerned with the origins of Perron’s Theorem in his masterful work on ordinary and generalized
continued fractions (1907) and its role in inspiring the remarkable work of Frobenius on nonnegative matrices (1912) that
produced, inter alia, the Perron–Frobenius Theorem. The paper is not at all intended exclusively for readers with expertise
in the theory of nonnegative matrices. Anyone with a basic grounding in linear algebra should be able to read this article
and come away with a good understanding of the Perron–Frobenius Theorem as well as its historical origins. The final section
of the paper considers the first major application of the Perron–Frobenius Theorem, namely, to the theory of Markov chains.
When he introduced the eponymous chains in 1908, Markov adumbrated several key notions and results of the Perron–Frobenius
theory albeit within the much simpler context of stochastic matrices; but it was by means of Frobenius’ 1912 paper that the
linear algebraic foundations of Markov’s theory for nonpositive stochastic matrices were first established by R. Von Mises
and V.I. Romanovsky. |
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