Eigenvalues and Eigenvectors

From left: Xining Zhang, Peter Denton and Stephen Parke
in front of the formula they discovered.

There was a very interesting story in Quanta Magazine recently titled Neutrinos Lead to Unexpected Discovery in Basic Math (November 13th 2019). The article begins:

After breakfast one morning in August, the mathematician Terence Tao opened an email from three physicists he didn’t know. The trio explained that they’d stumbled across a simple formula that, if true, established an unexpected relationship between some of the most basic and important objects in linear algebra. 

The formula “looked too good to be true,” said Tao, who is a professor at the University of California, Los Angeles, a Fields medalist, and one of the world’s leading mathematicians. “Something this short and simple — it should have been in textbooks already,” he said. “So my first thought was, no, this can’t be true.” 

Then he thought about it some more. 

The physicists — Stephen Parke of Fermi National Accelerator Laboratory, Xining Zhang of the University of Chicago and Peter Denton of Brookhaven National Laboratory — had arrived at the mathematical identity about two months earlier while grappling with the strange behavior of particles called 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 realised 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. 

Although the physicists could hardly believe they’d discovered a new fact about such bedrock math, they couldn’t find the relationship in any books or papers. So they took a chance and contacted Tao, despite a note on his website warning against such entreaties. 

“To our surprise, he replied in under two hours saying he’d never seen this before,” Parke said. Tao’s reply also included three independent proofs of the identity.

There's a nice graphic in the article. This is shown in Figure 1. There is an accompanying commentary:

Figure 1

Eigenvectors and eigenvalues are ubiquitous because they characterise linear transformations: operations that stretch, squeeze, rotate or otherwise change all parts of an object in the same way. These transformations are represented by rectangular arrays of numbers called matrices. One matrix might rotate an object by 90 degrees; another might flip it upside down and shrink it in half. 

Matrices do this by changing an object’s “vectors” — mathematical arrows that point to each physical location in an object. A matrix’s eigenvectors — “own vectors” in German — are those vectors that stay aligned in the same direction when the matrix is applied. Take, for example, the matrix that rotates things by 90 degrees around the x-axis: The eigenvectors lie along the x-axis itself, since points falling along this line don’t rotate, even as everything rotates around them. 

A related matrix might rotate objects around the x-axis and also shrink them in half. How much a matrix stretches or squeezes its eigenvectors is given by the corresponding eigenvalue, in this case ½. If an eigenvector doesn’t change at all, the eigenvalue is 1. 

Eigenvectors and eigenvalues are independent, and normally they must be calculated separately starting from the rows and columns of the matrix itself. College students learn how to do this for simple matrices. But the new formula differs from existing methods. “What is remarkable about this identity is that at no point do you ever actually need to know any of the entries of the matrix to work out anything,” said Tao. 

The identity applies to “Hermitian” matrices, which transform eigenvectors by real amounts (as opposed to those that involve imaginary numbers), and which thus apply in real-world situations. The formula expresses each eigenvector of a Hermitian matrix in terms of the matrix’s eigenvalues and those of the “minor matrix,” a smaller matrix formed by deleting a row and column of the original one.

All this got me thinking about eigenvalues and eigenvectors again, and more generally linear algebra. I was reminded of 3BLUE1BROWN's excellent 15-part series of YouTube videos on linear algebra. Here is number 14 on eigenvectors and eigenvalues:

I've watched most of the videos before but it's probably time to watch the whole series again, just to refresh my understanding of the topic. I also need to spend some time familiarising myself with how SageMath implements the calculation of eigenvectors. Figure 2 shows a screenshot of a SageMathCell calculations of the eigenvectors and eigenvalues for the matrix shown in Figure 1.

Figure 2

In the SageMath code, I don't understand the difference between the eigenvectors_right() versus eigenvectors_left() commands. They give the same result in this case. The eigenvectors are also displayed in a rather odd and difficult to read format but it's all there so at least I know how to do that. 

What's clear is that, by using \((1, 1, 0) \), \((1, -1, 0) \) and \((0, 0, 1) \) as basis vectors for the 3 dimensional space, the linear transformation described in Figure 1 will only have a scalar effect of these vectors. Specifically, it will have cause them to be

• unaltered along the \( (1, 1, 0) \) vector axis because eigenvalue is 1
• flipped along the \( (1, -1, 0) \) vector axis because eigenvalue is -1
• stretched by a factor of 2 along the \( 0, 0, 1) \) vector axis because eigenvalue is 2

In the old vector space with basis vectors [1, 0, 0], [0, 1, 0] and [0, 0, 1], the linear transformation in Figure 1 did have a scalar effect on the vector [0, 0, 1] by doubling it to [0, 0, 2] but the other two vectors ended up swapping places. Using the eigenvectors associated with the transformation as the basis vectors ensures that this sort of disruption is avoided.

After all of this, the details of the newly discovered identity have not been addressed really, apart from the photo showing the three discoverers in front of a blackboard where some of the details are discernible. Here is a link to Terence Tao's WordPress blog in which he discusses the new discovery. Figure 3 shows the newly developed theorem:

Figure 3: the new theorem as it appears on Terence Tao's blog

This is out of my league really but at least the article motivated me to revise some of the basics of linear algebra, so that's a good thing.

Read about Terence Tao's latest discovery regarding the Collatz Conjecture: 

https://t.co/h8cMC9QKes.