# The history and motivation of eigenvectors

I want to understand more about the history of eigenvectors. Was the discovery of eigenvectors inspired from an application to achieve a result in a historical context, was there a phenomenon which operated in some related way that was observed, was it discovered from the formulation and explored and mentioned till applications were found, or from an indirect usage in some other area of mathematics for solving equations?

• Did they solve a problem which was not solvable previously?
• Where can someone read the original track of their derivation from basic principles?
• Who were the main mathematicians to contribute to the development and use of eigenvectors and in which texts are they described?
• What does the Wikipedia article not tell you that is important here? Aug 18, 2015 at 17:49
• @HDE226868, The history section puts in the names and headers of the topics but does not give the original example contexts; looking at the uses the links you do not find the eigenvector history explained. The rest is text book basic examples and further applications; which are all great. But the construction of the eigenvector from the basic principles in the historical context is not elaborated.
– Vass
Aug 18, 2015 at 19:03

Eigenvectors (but not the word for them!) gradually appeared in 18s century in solving differential equations which we write now as $y'=Ay$ describing all sorts of oscillatory phenomena in the nature (mechanical vibrations, light, sound, etc.) Of course this was long before the words "matrix" and "vector" appeared. The simplest of these equations is $y''+k^2y=0$, it was completely understood by Huygens and its general solution is $y(t)=c_1\sin kt+c_2\cos kt$. Now we state this as "the operator of second derivative has eigenvalues $-k^2$ and eigenfunctions $\cos kt,\sin kt$. It may sound strange but eigenvectors in infinite dimensional spaces (eigenfunctions) appeared under various names long before linear algebra, and before the word "vector" came into common use. They played central role in the theory of small oscillations, and in Fourier's work on partial differential equations.

The words "eigenvector" and "eigenfunction" were not standard until well into 20s century (all kinds of other words were used like proper vectors characteristic vectors, etc.). The term "eigenvector" comes from German, I think under the influence of the book of Hilbert-Courant, first edition in 1920s.

• Indeed, when I first heard the fact that infinite-dimensional "linear algebra" preceded finite-dimensional (e.g., Sturm-Liouville... Hilbert-Schmidt...) I was fairly amazed! :) Apparently "matrices" were just used to track determinants, as a part of nascent "invariant theory", in most of the 19th century (Sylvester, Cayley, et al). Dec 9, 2023 at 15:57
• Terminology. As late as 1958, Halmos (in his text Finite-Dimensional Vector Spaces) proposed use of the terms translated into English: proper value, proper vector. Dec 9, 2023 at 16:37
• @Gerald Edgar: Halmos' proposal seems very reasonable but it did not take root. All undergraduate textbooks use the ugly German-English word eigenvalue. Dec 10, 2023 at 4:37

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".

• @MichaelE2 I have updated the link to the one you pointed out.
– KCd
Dec 15, 2019 at 3:50