Eigenvectors and eigenvalues are independent, and must normally be calculated separately from the rows and columns of the matrix itself. College students learn how to do this with simple matrices. But the new formula is different from the existing methods.
The formula discovered by the three physicists expresses each eigenvector matrix of the Hermitian matrix in terms of its eigenvalues and those of the “minor matrix,” a smaller matrix formed by deleting the row and column of the original matrix.
As if the calculations weren’t hard enough already, a bizarre effect first identified by the physicist Lincoln Wolfenstein in 1978 makes the neutrino matrix even more nightmarish. Neutrinos seldom interact with matter in the usual sense, but Wolfenstein realized that passing through matter rather than empty space nevertheless changes the way neutrinos propagate. As an electron neutrino zooms through matter, it will occasionally interact with an electron in an atom, effectively swapping places with it: The electron neutrino transforms into an electron and vice versa.
Such swaps introduce a new term in the matrix that affects the neutrinos of electrons, which greatly complicates mathematics. It is this “Wolfenstein Matter Effect” that pushed Parke, Zhang and Denton to look for a way to simplify the calculations.
The expressions for the own values are simpler than those of the eigenvectors, so Parke, Zhang and Denton started out there. Previously, they had devised a new method to closely approximate their eigenvalues. With these in hand, they noticed that the long proprietary expressions seen in previous works were equal to the combinations of those eigenvalues.
In fact, there was a similar equation, but it had gone unnoticed because it was in disguise.
εnigma 
Reblogged this on εnigma.
LikeLike