June 2, 2020

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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 twox+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.