(Re)Discovering Identities

In November, I ran across a very interesting article in QuantaMagazine “Neutrinos Lead to Unexpected Discovery in Basic Math by Natalie Wolchover. She described the discovery that three physicists — Stephen Parke (Fermi National Accelerator Laboratory), Xining Zhang (University of Chicago) and Peter Denton (Brookhaven National Laboratory) had made about eigenvalues and eigenvectors while studying neutrinos.

“They’d noticed that hard-to-compute terms called “eigenvectors,” describing, in this case, the ways that neutrinos propagate through matter, were equal to combinations of terms called “eigenvalues,” which are far easier to compute. Moreover, they realized that the relationship between eigenvectors and eigenvalues — ubiquitous objects in math, physics and engineering that have been studied since the 18th century — seemed to hold more generally.” – Natalie Wolchover

Neutrinos, which are sub-particles that interact with matter weakly, have oscillations between different types that can be described by eigenvalues and eigenvectors. In particular,

“The mathematical description of how neutrinos interact with matter involves square arrays of numbers called matrices. Every matrix has a set of characteristic numbers called eigenvalues; and along with each eigenvalue goes a direction in space called an eigenvector.” – Peter Lynch, “Particle physics gives maths potentially powerful new tool”

With the help of Terrence Tao and Van Vu, the Eigenvector-eigenvalue identity (shown in Figure 1) was proven! Formulas to convert from eigenvalues to eigenvectors exist but they can be hard to compute. This “new” identity states that if you have a square Hermitian matrix (such as the matrix associated with neutrino scattering), one can relate its eigenvalues to its eigenvectors through the eigenvalues of its minors (i.e. a sub-matrix of the original matrix with some rows and columns removed).

Figure 1. From Terrence Tao’s blog post.

While the discovery was exciting on its own, they also noticed that similar versions of this identity were independently discovered by others after this article was published. In his blog post, “Eigenvectors from eigenvalues: A survey of basic identities in linear algebra”, Tao describes,

“Within a few weeks we became informed (through private communication, online discussion, and exploration of the citation tree around the references we were alerted to) of over three dozen places where the identity, or some other closely related identity, had previously appeared in the literature, in such areas as numerical linear algebra, various aspects of graph theory (graph reconstruction, chemical graph theory, and walks on graphs), inverse eigenvalue problems, random matrix theory, and neutrino physics.” – Terrence Tao

For example,  What I love about this article is how it portrays two ideas that resonate with my experience in mathematics:

(1) Mathematics is full of discovery (and rediscovery) even when studying well-known objects.

(2) Nature is indeed written in the language of mathematics and often mathematics is also written in the language of nature.

As Mike Lawler describes in his blog, “Sharing the Eigenvectors from Eigenvalues paper with my son”, this discovery is also a nice way for those learning linear algebra to play with a result that relates to what they cover in class.

“I think this new paper is an incredible lucky break for anyone teaching linear algebra now or in the future. It really isn’t that often that a new math paper has a result that is accessible to young students.” – Mike Lawler

An advantage of being a mathematician is that the more that you immerse yourself in math the more you see it all around you. At least for me, even the most personal self-reflections have a math flare to them. As a new decade approaches, I’ve seen a lot of Twitter posts reflecting on what they have accomplished and how they’ve changed over the last decade. This made me ask myself, “how has my experience of mathematics changed during the last 10 years?”

Back in 2009, I was a sophomore barely starting to learn the vastness of math. I would go to the library and promise myself I would come back one day and be able to read any math book in the stacks. I used to think mathematics as a sorta abstract and above all else. When I started to meet other mathematicians and learned of their experiences, how they played with concepts, and their own passion for the subject, mathematics became much more than ideas, it became about the people and the world around me too. Fast forward to now, I’ve realized how naive I was and what a beautiful dream that was. In the video The Map of Mathematics”,  Dominic Walliman captures the richness and evolution of math and as he shares,

“Now the thing I have loved most about learning maths is the feeling you get where something that seems confusing finally clicks in your brain and everything makes sense. Like an epiphany, kind of like seeing to the matrix” – Dominic Walliman

And it clicked,  I started thinking about matrices, and in particular, eigenvalues and eigenvectors. Historically, the prefix eigen, has been a cause of debate. As described in Math Origins: Eigenvalues and Eigenvectors by Erik K. Tou,

“By the middle of the 20th century, there were at least five different adjectives that could be used to refer to the solutions in our particular type of matrix equation: secular, characteristic, latent, eigen, and proper. In general, though, two naming conventions dominated: eigen- (from Hilbert’s German writings) and characteristic/proper (from Cauchy’s French writings and von Neumann’s translation of eigen-). In the United States, the peculiar prefix eigen– won the debate.” – Erik K. Tou

Even in many languages, the idea behind this concept remains the same: eigenvalues (and eigenvectors) tell us something about ourselves and definitely as this decade ends I will be reflecting on my own eigenvalues, maybe I’ll also rediscover new identities.

Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)


About Vanessa Rivera-Quinones

Mathematics Ph.D. with a passion for telling stories through numbers using mathematical models, data science, science communication, and education. Follow her on Twitter: @VRiveraQPhD.
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