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What is an eigenvalue?
21–30 of 167 posts
Re: What is an eigenvalue?
#22Does 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.
Moment of inertia tensor [1], and principal axes of rotation [2].
Principle axes are the axes where a weightless body can rotate around without "wobbling". These axes are orthogonal to each other. If a rigid body has I_1 [1] https://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_tens...
[2] https://en.wikipedia.org/wiki/Moment_of_inertia#Principal_ax...
Re: What is an eigenvalue?
#23Does 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.
Markov probability matrix where the entries are probabilities of some physical event happening.
The the eigenvectors will be the long term stable state probabilities.
Re: What is an eigenvalue?
#24I do think, the term eigenvalue is rather opaque, and should be replaced by a more plain-english terminology that readily conveys its meaning.
As a native German speaker, I don't understand your problem. It's very much not opaque, plain terminology that easily conveys meaning. ;-)
Perhaps, we should compromise and name it after Leonhard Euler? That should clear up the confusion.
Re: What is an eigenvalue?
#25Earlier quoted context omitted.
Take a linear map from some space to itself, and ask: What lines (through the origin) are mapped back to themselves? Those are the eigenvectors, and the amount by which they're elongated or shortened are the eigenvalues. So, if we talk about 3d space, and we rotate things - the rotation axis is unchanged. That's an eigenvector (with eigenvalue 1). If we mirror things - any vector in the mirror plane remains unchanged…
What does 'through the origin' mean in a physical system?
It means it doesn't matter where it is: you can choose the origin, ie the point you measure from, it is arbitrary. Or another way of saying that is you can move the system to a different set of coordinates and it works in the same way.
... which means it's probably an imaginary physical system.
Maybe a good physical example is a piece of cloth that warps in 2D, and shrinks, when washed? Eigenvectors would describe the warping (skew, say) and eigenvalues the shrinkage relative to the original warp and weft.
Steve Brunton on YouTube has really good videos on eigenvectors & eigenvalues in context of matrix algebra (and then applied to simultaneous differential equations); https://youtube.com/watch?v=ZSGrJBS_qtc .
Re: What is an eigenvalue?
#26Does 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.
Take an airplane... it’s dynamics are described by a series of differential equations. We want to know if it’s stable! If the wife values of the dynamics are real and greater than 1 it’s unstable. If the eigenvalues are complex and have a modulus greater than 1 it will oscillate instability. If one is equal to one, it will cause everyone to vomit.
Re: What is an eigenvalue?
#27Does 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.
I read an interview with Australian wire music composer Alan Lamb that a stringed instrument with multiple overtones vibrating on the string can be analyzed by breaking down the vibration into eigenvalues, but I've never found any reference material that explain that. I'm wondering if he was referring to FFT.
Complex exponentials are the eigenfunctions of the Fourier transform. In other words, frequency component values are the eigenvalues.
https://en.m.wikipedia.org/wiki/Eigenfunction#Vibrating_stri...
Re: What is an eigenvalue?
#28For a completely different approach, see this answer:
https://www.reddit.com/r/explainlikeimfive/comments/1avwm7/c...
Re: What is an eigenvalue?
#29Does 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.
Also in computer science.. a web sites page rank is the eigenvalue of the connectivity matrix.
Re: What is an eigenvalue?
#30I do think, the term eigenvalue is rather opaque, and should be replaced by a more plain-english terminology that readily conveys its meaning.
“Characteristic value.” I guess that’s a little better actually.