Eigenvalues

An eigenvector is simply a vector that is unaffected (to within a scalar value) by a transformation. Formally, an eigenvector is any vector x such that for an operator Ω, Ωx=λx for some scalar constant λ

All operators of dimension n have exactly n
eigenvectors/eigenvalues (though these are only all distinct if Ω is diagonalizable).

Generalizing the idea of an eigenvector to any thing that is affected only up to a scalar value by some operator, here are a few examples:

• In math, the set of exponential functions (e.g. nx) are the eigenfunctions of the differentiation operator, and ex

is the eigenfunction with eigenvalue 1. You can use eigenvector/value completeness in the differentiation operator to prove Euler's identity, that eiθ=cosθ+isinθ

, and to prove that ex

corresponds to its Taylor series.

• In quantum mechanics, an "eigenstate" of an operator is a state that will yield a certain value when the operator is measured. The eigenvalues of each eigenstate correspond to the allowable values of the quantity being measured. For example, the energy eigenstates of an electron in a hydrogen atom (a simple harmonic oscillator), corresponding to energies of En=−Ry/n2

will always give their corresponding energies if their energies are measured. However, a state composed of a linear combination of eigenstates, such as |ψ⟩=12√|0⟩+12√|1⟩
will give energies of either E0
or E1
with equal probabilities. In general, a state |ψ⟩
composed of a linear combination of eigenfunctions ψn
of any observable variable (energy, spin, momentum, etc.) given by

|ψ⟩=∑n=0Nan|ψn⟩ will always yield a measured value that is an eigenvalue of the observable, with the probability of each value being given by a2n .

• Also in quantum mechanics, two observables generate an uncertainty principle if they do not commute - meaning that their matrix representations are not simultaneously diagonalizable, so they don't share a set of eigenvectors/eigenvalues. The most common example for this is the Heisenberg uncertainty principle, given by σx⋅σp≥ℏ2

, since and p

do not commute and thus do not share a set of common eigenstates. However, there are an infinite number of these corresponding uncertainty principles, each corresponding to some set of incompatible physical quantities.

• Google's PageRank algorithm generates transition probabilities between webpages by looking at the number of links from each page to each other page, and makes a giant "matrix" from these values. The resulting eigenvectors and eigenvalues of this "operator" provide a very reliable metric for ranking pages by relevance and quality of content.

• The first generations of airplanes were prone to disintegrate mid-flight due to a phenomenon called "flutter", which is when turbulence from air passing over the plane drives the plane at its natural resonance frequencies (the "eigenfrequencies", or vibrational modes, of the object). Modeling the aircraft and computing the eigenfrequencies allows engineers to modify the design accordingly, which usually involves using slightly different materials with nearly relatively prime resonant frequencies in key parts of the aircraft.

• There are many other applications, but these are just a few that come to mind. Many problems across disciplines are actually eigenvalue problems in disguise. https://physics.stackexchange.com/questions/233922/eigenvalue-physical-meaning#:~:text=In%20quantum%20mechanics%2C%20an%20%22eigenstate%22%20of%20an%20operator,the%20allowable%20values%20of%20the%20quantity%20being%20measured.