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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机译：非负矩阵理论是起源于纯数学问题的理论的一个示例，后来被证明在概率论，数值分析，经济学，动力学规划和人口统计学等不同领域具有广泛的应用。该理论的核心是通常所说的Perron-Frobenius定理。它是由Oskar Perron关于正矩阵的定理（通常称为Perron定理）启发而来的。本文主要涉及佩伦定理在其关于普通和广义连续分数的出色著作（1907年）中的起源，以及其在激发弗罗贝尼乌斯在非负矩阵上的杰出著作（1912年）中的作用，该著作除其他外还产生了佩隆-弗罗贝尼乌斯定理。本文绝不完全面向具有非负矩阵理论知识的读者。任何具有线性代数基础的人都应该能够阅读本文，并对Perron-Frobenius定理及其历史渊源有很好的理解。本文的最后一部分考虑了Perron-Frobenius定理的第一个主要应用，即在马尔可夫链理论中的应用。 1908年，马尔科夫（Markov）引入同义链时，尽管在更简单的随机矩阵环境下，还是对佩隆-弗罗贝尼乌斯（Perron-Frobenius）理论的几个主要概念和结果都表示赞同。但是正是通过Frobenius的1912年论文，才由R. Von Mises和V.I.首次建立了马尔可夫非正定随机矩阵理论的线性代数基础。罗曼诺夫斯基。
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