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He admits to being good at solving puzzles. In fact, he is credited with co-discovering another important pattern in that has streamlined particle physics calculations and inspired discoveries ever since.
Still, the fact that the strange behavior of neutrinos could lead to new insights about matrices came as a shock. In fact, a similar formula did already exist, but it had gone unnoticed because it was in disguise. In September, Tao got another out-of-the-blue email, this time from Jiyuan Zhang, a mathematics graduate student at the University of Melbourne in Australia.
Zhang and Forrester were working in an area of pure mathematics called random matrix theory. They had applied the formula in their study of the randomized Horn problem — a problem connected to one that Tao and a colleague solved in In an email to Quanta , Forrester explained that the formula first appeared in yet another form in a paper by Yuliy Baryshnikov , a mathematician now at the University of Illinois, Urbana-Champaign, whose work Forrester and Zhang had built on.
And people are just beginning to look in the right places. Update on December 4, In the weeks following the publication of this article, the researchers became aware of over three dozen places where the identity had appeared in the literature since They have now rewritten their original paper to include the history of the identity, along with all seven known proofs.
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And while these variations have disappeared from common use in the United States, we'll see that even today the argument is not quite over. Before going further, though, note that the prefix eigen- predates its mathematical use. Old English used the word agen to mean "owned or possessed by ," and while this usage no longer exists in modern English, eigen is used to mean "self" in modern German.
For example, the word eigenkapital is usually translated into English as equity , though a more literal translation would be "self capital. So we can see that the prefix eigen - had a well-established usage in German and once upon a time, in English. How, then, did it get attached to mathematics?
At this time, astronomers and mathematicians were making detailed observations of the planets in order to validate the mathematical models inherited from Kepler and Newton's laws of motion. One famous outcome of this program was Urbain Le Verrier 's prediction of the existence of Neptune based on observed perturbations in the orbit of Uranus. In celestial mechanics, most perturbations are periodic; for example, Neptune's influence on Uranus's orbit waxes and wanes as the planets revolve around the sun.
However, one perplexing issue in the 18th and 19th centuries was that of secular perturbations—those non-periodic perturbations that increase gradually over time. Today, the word secular usually refers to those aspects of society and culture that are not religious or spiritual in nature.
However, it also carries the meaning of something whose existence persists over long stretches of time coming from the Latin saecularis , "occurring once in an age" , which is the more accurate usage in this case. As it happens, Cauchy's original interest in this subject was not secular perturbation, but rather the axes of motion for certain surfaces in three dimensional space.
In an note to the Paris Academy of Sciences, Cauchy wrote:. It is known that the determination of the axes of a surface of the second degree or of the principal axes and moments of inertia of a solid body depend on an equation of the third degree, the three roots of which are necessarily real.
However, geometers have succeeded in demonstrating the reality of the three roots by indirect means only The question that I proposed to myself consists in establishing the reality of the roots directly In today's language, we would say that Cauchy's research program was to show that a symmetric matrix has real eigenvalues. For simplicity, we will only consider the three-dimensional version of the problem.
Figure 1. Readers might recognize this as the Lagrange multiplier method found in most multivariable calculus courses. Figure 2. Cauchy reasoned that, if the characteristic polynomial had complex roots, they must come in conjugate pairs. From here, Cauchy showed that the product of these determinants must be zero, i. The interested reader should take a look at the original paper, available from Gallica.
While this paper was focused on pure mathematics, his ultimate goal was to solve physics problems: "It is the integration of linear equations , and above all linear equations with constant coefficients , that is required for the solution of a large number of problems in mathematical physics.
Figure 3. Cauchy's method for solving a system of linear, first-order differential equations with constant coefficients, equivalent to a modern-day eigenvalue problem. The history of eigenvectors is more or less the same as the history of eigenvalues.
I recommend looking at Steen's paper "Highlights in the history of spectral theory". Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. The history and motivation of eigenvectors Ask Question. Asked 6 years, 2 months ago. Active 1 year, 11 months ago. Viewed 4k times.
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