And though I...understand all mysteries, and all knowledge;....and have not Love, I am nothing.
1 Corinthians 13:2
"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.
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.
Eigenvectors and eigenvalues are ubiquitous because they characterize
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.
It’s unusual in mathematics for a
tool to appear that’s not already associated with a problem, he said.
But he thinks the relationship between eigenvectors and eigenvalues is
bound to matter."
Quanta Magazine/Natalie Wolchover
