A New Discovery About Matrices

The first time I studied linear algebra was when I was an exchange cadet at the Air Force Academy.  That was the only course in all of my college education in which everyone in the course failed.  Well, I earned a low 60-something percent, but since mine was the highest grade in the class, I got an A.  Yay me.

Not having learned anything in that course, and being petrified of linear algebra because of it (and perhaps because over 25 years had since passed), I convinced my advisor to let me retake the course in my master's program.  It was in that course that I learned a little bit about eigenvalues and eigenvectors, interesting components in the study of matrices.  It was when meeting a former student a few years ago, and telling him about what little we teach about matrices in pre-calculus class, that I learned from him about a YouTube video that explained eigenvalues.  That was when all those calculations I'd done made actual sense!

Physicists have now added to our understanding eigenvalues and eigenvectors:

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 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.

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.

A week and a half later, the physicists and Tao, whom Parke called “a fire hose of mathematics,” posted a paper online reporting the new formula. Their paper is now under review by Communications in Mathematical Physics. In a separate paper submitted to the Journal of High Energy Physics, Denton, Parke and Zhang used the formula to streamline the equations governing neutrinos.

Experts say that more applications might arise, since so many problems involve calculating eigenvectors and eigenvalues. “This is of really broad applicability,” said John Beacom, a particle physicist at Ohio State University. “Who knows what doors it will open"...

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.

So cool.