# The Theorem That Applies to Everything from Search Algorithms to Epidemiology

On a new episode of our podcast My Most loved Theorem, my cohost Kevin Knudson and I talked with Carina Curto, a mathematician at Penn State University who specializes on mathematics utilized to biology and neuroscience. You can pay attention to the episode right here or at kpknudson.com, where there is also a transcript.

Curto instructed us about the Perron-Frobenius theorem, which comes from the industry of linear algebra. As Curto gushes in the episode, linear algebra is just one of the workhorses of mathematics. It delivers a robust set of techniques for modeling and knowledge systems of equations that come up in many diverse contexts.

A person of the central objects of study in linear algebra is a matrix. A matrix is an array of numbers, but it is much more than that. The numbers are arranged in rows and columns and are generally assumed of as a shorthand representation for a transformation of a house. For instance, just one way to transform the airplane is to take just about every level in the airplane, created as (*x,y*) and send the to start with coordinate, *x*, to the sum two*x*+*y* and the next coordinate, *y*, to *x*+*y*. The matrix

encodes that transformation.

In common, a ray of factors emanating from the origin can be stretched and rotated by a transformation encoded by a matrix, but in a special route called the eigenvector, the transformation is confined to stretching or shrinking. There is no rotation. The amount of money by which a device duration phase pointing in this route is stretched or shrunk is the eigenvalue. The Perron-Frobenius theorem states that for a sq. matrix with all positive entries, there is a one of a kind premier true eigenvalue and that its corresponding eigenvector has positive *x* and *y* coordinates.

I vaguely remembered listening to about the Perron-Frobenius theorem in advance of conversing with Curto, but I did not at first uncover it significantly powerful. I uncovered early in my to start with linear algebra course not to set too much inventory into the numbers you see in a matrix. That is, you should not consider you can fully grasp what a matrix will do based mostly only on what its entries seem like. I consider component of my failure to recognize the Perron-Frobenius theorem before is that I was a small little bit suspicious of a theorem that did allow for me to know a little something about a transformation based mostly on the numbers in a matrix. And to top rated it off, not many matrices have all positive entries, so what is the level of a theorem that applies to these kinds of an exceptional set of matrices?

Curto certain me that the theorem is more helpful than I gave it credit score for. Accurate, there is no explanation to consider any arbitrary matrix is heading to have all-positive entries, but many purposes do offer principally with all-positive matrices. The matrices that signify functions connected to populace dynamics, demographics, economics, and internet lookup algorithms typically have all-positive entries, so the Perron-Frobenius theorem applies to them. For the present-day instant, there is even an epidemic model, the Kermack-McKendrick model, that uses the Perron-Frobenius theorem. (My point out of this model must not be construed as an encouragement to take up armchair epidemiology.) Readers and listeners who want a more considerable introduction to the many proofs and purposes of the Perron-Frobenius theorem must examine out a paper called, correctly ample, The A lot of Proofs and Applications of the Perron-Frobenius Theorem.