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What is an eigenvalue?

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Re: What is an eigenvalue?

#132

Post by patrick451 »

kiernanmcgowan wrote: post #114

Eigen-things can also be thought of "fixed values" of a "thing" transformation. For example - the eigenfunction of a derivative is e^x since when you run the derivative function on e^x you get.... e^x

Another good example is that sinusoids are eigenfunction of a linear dynamical system. If you drive the system with a sinusoid, the output is also a sinusoid of the same frequency, but different amplitude and phase. In other words, the frequency response is the eigenvalue at a given frequency.

Re: What is an eigenvalue?

#133

Post by pictureofabear »

Waterluvian wrote: post #48
martin_balsam wrote: post #46

Earlier quoted context omitted.

But it is named after its characteristic, albeit in German

Well… boy did I pick the wrong example to bring this up with. Alas, I’ll leave my shame here for all to see.

Your point is well taken on many examples, but this was not one. hah

Re: What is an eigenvalue?

#135

Post by c-baby »

lp251 wrote: post #127
c-baby wrote: post #105

Earlier quoted context omitted.

5z + 2 is linear?

affine, not linear. describes a line that doesn't go through the origin. that pesky shift breaks linearity 5(2z) + 2 != 2(5z + 2)

Good point. But what makes studying these functions interesting? Like what key theorems govern this class of functions?

Re: What is an eigenvalue?

#137

Post by j2kun »

AdamH12113 wrote: post #2

Does anyone know of an example of a simple physical system where eigenvalues have a physical interpretation? The examples I know of are all in quantum mechanics, which is a bit abstract for me.

Here's one: https://pimbook.org/demo/index.html

Eigenvectors are the fundamental frequencies of a spring system

Re: What is an eigenvalue?

#138

Post by hither_shores »

adamnemecek wrote: post #60

All these points fail to mention that they are fundamentally self-relationship Lawvere's fixed point theorem is I think the best formulation of the idea https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore... I've been putting together a brain dump on the topic https://github.com/adamnemecek/adjoint/ Join the discord https://discord.gg/mr9TAhpyBW

There's not really a connection here, the category of vector spaces isn't cartesian closed.

Re: What is an eigenvalue?

#139

Post by monktastic1 »

kiernanmcgowan wrote: post #114

Eigen-things can also be thought of "fixed values" of a "thing" transformation. For example - the eigenfunction of a derivative is e^x since when you run the derivative function on e^x you get.... e^x

Functions satisfy the laws of a vector space (well, certain classes of functions, such as L^2 anyway -- I'm certainly no expert here). I.e., functions are vectors. And the derivative is a linear operator on that space: d/dx(af + bg) = ad/dx(f) + bd/dx(g). Therefore d/dx has eigenvalues, one of which you properly identified as e^x (and more generally, e^ax, as others have pointed out).

Re: What is an eigenvalue?

#140

Post by adamnemecek »

hither_shores wrote: post #138
adamnemecek wrote: post #60

All these points fail to mention that they are fundamentally self-relationship Lawvere's fixed point theorem is I think the best formulation of the idea https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore... I've been putting together a brain dump on the topic https://github.com/adamnemecek/adjoint/ Join the discord https://discord.gg/mr9TAhpyBW

There's not really a connection here, the category of vector spaces isn't cartesian closed.

The category of Chu spaces is through.
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