The essence of the eigen problem

[Carbon273]

TL;DR Summary

I am a bit doubtful, is the eigen problem only applicable for linear scenarios? Or are they also applicable for nonlinear scenarios. In linear algebra we are given the equation, Ax=λx. I understand this equation only works if A is a linear transformation of the vector x. Which as a result, produces a scaled value of the vector x. The linear transformation would have to abide by the fact that the origin is in place and that the grid lines are parallel. In reference to a blueBrown video:

This is with respect to the vector x.

 

[FactChecker]

Nonlinear problems are often analyzed locally by "linearizing" the problem. Then the eigenstructure indicates the local behavior of the system in that region of the state space.

 

[fresh_42]

Summary: I am a bit doubtful, is the eigen problem only applicable for linear scenarios?

In the non linear case it is called fixed point problem: $f(x)=x$ or in case of dynamical systems, an attractor. So yes, eigen implies linearity.

 

[S.G. Janssens]

In the non linear case it is called fixed point problem: $f(x)=x$ or in case of dynamical systems, an attractor.

The existence of a fixed point and whether or not it is attracting are two different things.

So yes, eigen implies linearity.

No, it doesn't, there exists a large literature on nonlinear eigenvalue problems and they are not just fixed-point problems going by a different name.

 

[fresh_42]

The existence of a fixed point and whether or not it is attracting are two different things.

I know, but they are related: $=1$ vs. $<1$.

No, it doesn't, there exists a large literature on nonlinear eigenvalue problems and they are not just fixed-point problems going by a different name.

Define a nonlinear eigenvalue! I want to see where this is not a fixed point. Even the linear case is a fixed point problem.

 

[Carbon273]

OK guys which is it lol? Maybe particularly focus on the eigen problem with respect to x. Also I would love to see a nonlinear eigen problem.

 

[fresh_42]

OK guys which is it lol? Maybe particularly focus on the eigen problem with respect to x. Also I would love to see a nonlinear eigen problem.

Here is an example of "the large literature", but it is close to the linear world, where it arose from as a generalization. I still wouldn't use the prefix eigen in nonlinear cases, except in physics where we call the frequencies in normal modes eigen-frequency. If you use it outside of linearity you will have to tell precisely what is meant. Without further specifications it is linear.

 

[WWGD]

Here is an example of "the large literature", but it is close to the linear world, where it arose from as a generalization. I still wouldn't use the prefix eigen in nonlinear cases, except in physics where we call the frequencies in normal modes eigen-frequency. If you use it outside of linearity you will have to tell precisely what is meant. Without further specifications it is linear.

Well, doesn't "Eigen" mean singular in English? What singularity are we then addressing?

 

[fresh_42]

Well, doesn't "Eigen" mean singular in English? What singularity are we then addressing?

Eigen cannot be properly translated, that's why it migrated. They have tried proper but proper has too many different meanings already. Anyway, it would be the closest word in the sense of eigen- as in Eigenschaft = property as in a description, not as in an ownership, which would be property = Eigentum. See, proper is pretty overloaded in English, and eigen has the advantage, that it can easily be spoken in English, once you accepted that ei = i and g = gh.

 

[WWGD]

Eigen cannot be properly translated, that's why it migrated. They have tried proper but proper has too many different meanings already. Anyway, it would be the closest word in the sense of eigen- as in Eigenschaft = property as in a description, not as in an ownership, which would be property = Eigentum. See, proper is pretty overloaded in English, and eigen has the advantage, that it can easily be spoken in English, once you accepted that ei = i and g = gh.

Well, overloaded but not as much as the word 'Normal' , which I think has some 20 different meanings in Mathematics.

 

[aNewDawn]

In "No Bullshit Guide to Linear Algebra", Ivan Savov writes, «The theorem [Fundamental Theorem of Algebra] is "fundamental" because it tells you we won't ever need to invent any "fancier" set of numbers to solve polynomial equations.»

So I suggest that a better translation of "eigen" (German) into English be "fundamental".

• eigenfunction ----> fundamental function
• eigenvalue ----> fundamental value

There is no "fancier" (simpler?) function, together with some associated value(s), that can be written. THAT'S IT! The fundamental function and it's associated fundamental value are, well, FUNDAMENTAL. Digging deeper for for something something "more fundamental" is a fool's quest.

 

[HallsofIvy]

But the German word "eigen" is best translated as "proper" or "characteristic", not "fundamental".

 

[marcusl]

Proper and characteristic were in common use in the English math/physics literature until quantum mechanics was developed in the 1920's and German terms took over. We still use the term "characteristic equation" or "secular equation" for the equation whose roots give the characteristic (eigen) values.

 

[fresh_42]

We have had a similar discussion a while back:

"At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.

J. Dieudonné mentioned in his "Abrégé d'histoire des mathématiques 1700-1900" that first examinations of determinants and eigenvalues, although not named as such, date back to Lagrange's work on partial differential equations (1762) and J.C.F. Sturm - J. Liouvilles work on operators in the first half of the 19th century..

https://www.physicsforums.com/threads/where-does-the-nomenclature-eigen-originate.843038/

 

[marcusl]

Thanks, it’s nice to see that history.

 

[Stephen Tashi]

Summary: I am a bit doubtful, is the eigen problem only applicable for linear scenarios?

I'm curious whether you think "linear scenarios" is a very limited set of situations. It includes linear transformations and there are many linear transformation besides those represented by $n$ by $m$ matrices. For example, let $g(x)$ be a given function. Define the transformation $T_g$ that maps a function to another function by: $T_g( f(x)) = f(g(x))$

The eigen problem for $T_g$ is to find functions $h(x)$ such that $T_g(h(x)) = \lambda h(x)$ for some number $\lambda$.

As another example, define the transformation $D$ that maps functions to functions by $D(f(x)) = f'(x)$ You probably know some solutions to the eigen problem for $D$.

 

[atyy]

Well, doesn't "Eigen" mean singular in English? What singularity are we then addressing?

"Eigen" means "own" in the sense the eignvalues of an operator are owned by the operator, or a characteristic of the operator. So eigenvalues have also been called characteristic values.

 

[S.G. Janssens]

What singularity are we then addressing?

In the linear setting, if $A$ is a matrix in $\mathbb{C}^{n \times n}$, then the resolvent map $z \mapsto (z I - A)^{-1}$ is meromorphic, takes values in $C^{n \times n}$ and its isolated singularities are of course precisely the eigenvalues of $A$. The matrix-valued Cauchy integral
$$
\frac{1}{2\pi i}\oint_{\Gamma_{\lambda}} (z I - A)^{-1}\,dz
$$
is just the spectral projector associated with the eigenvalue $\lambda$ encircled by the contour $\Gamma_{\lambda}$. (This can be generalized to apply to a closed linear operator $A$ on a complex Banach space. Then the above integral is named after Dunford.)

So, for me perhaps the "essence of the eigenproblem" is the investigation of singularities of classes of meromorphic matrix-valued (or: linear operator valued) functions. Depending on the functional form, these problems may then be called linear or nonlinear.