Perron's Theorem

Undergraduate Thesis for Colorado State University Honors Program

Natalie Burke, Fall 2020

In Fall 2020, I wrote my undergraduate thesis on Perron's Theorem. In my paper, I explain select versions of Perron's theorem in order of their creation. At the time, I hoped to reconfigure my work into a website that would be interactive and accessible to all. This is the result of that work.

In this webpage, I will explain Perron's Theorem and Perron-Frobenius theory from linear algebra. There is a page on the historical perspective of Perron-Frobenius theory. Also, there is a page where applications and examples are investigated, including applications to Markov chains, Leslie's population model, and Google's Page Rank algorithm. Then there are pages explaining key background mathematics and works cited respectively.

Perron's theorem was first stated and proved in 1907. Perron's theorem allows us to determine quite a few conclusions about positive matrices. It tells us that positive square matrices behave how one might intuitively expect in that they have at least one positive eigenvalue and positive eigenvector. Perron-Frobenius theory is an extension of this theorem on nonnegative matrices (Bernhardt).