Many problems involve calculating eigenvectors and eigenvalues.

What does this mean?

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.

Shape-Shifting Particles

That application would be neutrinos: the oddest, least understood, most reclusive of the known fundamental particles. Neutrinos pass through each of our bodies by the trillion each second, but because they barely register, many of their properties remain unknown.

Intriguingly, theory suggests that differences in the behavior of neutrinos and antineutrinos could be what allows matter to dominate over antimatter in the universe. If these opposites had arisen in equal amounts in the Big Bang, they would have mutually annihilated, yielding a cosmos empty of everything except light. A distinction between neutrinos and antineutrinos could be what allowed the all-important surplus of matter to accrue. “If they act differently, that will give us some hint as to why the universe is filled with matter,” said Deborah Harris, a physicist at York University and Fermilab who works on a neutrino experiment called DUNE (for Deep Underground Neutrino Experiment) aimed at measuring such differences.

The experiment, which will measure neutrinos shot from Fermilab in Illinois to an underground detector 1,300 kilometers away in South Dakota, makes use of the fact that neutrinos come in one of three possible “flavors” — electron, muon or tau. But each neutrino flavor is a quantum mechanical mixture, and neutrinos oscillate between flavors on the fly. As a neutrino from Fermilab travels along, its mixture changes, so that a muon neutrino might morph into an electron neutrino or a tau neutrino.

An egregiously complicated three-by-three matrix describes these oscillations. From the eigenvectors and eigenvalues, physicists can calculate an expression for the likelihood that a muon neutrino will oscillate into an electron neutrino by the time it reaches South Dakota. They can also calculate an expression for the probability that a muon antineutrino will become an electron antineutrino.