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What is an eigenvalue?
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Re: What is an eigenvalue?
#12Does 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.
Re: What is an eigenvalue?
#13Does 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.
Natural frequencies of mechanical systems are eigenvalues of it’s equation of motion.
Any references to help me unpack what you just said there?
Re: What is an eigenvalue?
#14Does 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.
Don't know if this counts as a "physical" system, but Google's original PageRank algorithm famously uses eigenvectors and eigenvalues: https://math.stackexchange.com/questions/936757/why-is-pager...
I have a brief overview of eigenvectors as a 2D shear transformation in this overview of PageRank:
https://web.archive.org/web/20130728183938/williamcotton.com...
Re: What is an eigenvalue?
#15Does 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.
If you discretize the string into a bunch of tiny masses, linked together by a bunch of tiny springs, you can build a mass Matrix (diagonal) M and a Stiffness matrix K (element ij = stiffness of spring that links mass I and mass j).
I can't remember the next part exactly, you can look it up in a textbook, but you multiply the matrices KMK, or similar, and the eigenvalues of this are the natural frequencies of the string. The eigenvectors represent the mode shapes, ie the displacement of each mass element.
The same technique is used in Finite Element Analysis to find the modes and modeshapes of complex structures (a car frame, a bridge, etc)
Re: What is an eigenvalue?
#16I do think, the term eigenvalue is rather opaque, and should be replaced by a more plain-english terminology that readily conveys its meaning.
Re: What is an eigenvalue?
#17Does 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.
Think of a fun house mirror that for the sake of this example make you look twice as tall but 20% skinnier. This can be modeled by a two-by-two matrix with eigenvalues of 2 and 0.8. (Indeed, it will have them on the diagonals which makes it easier to study.)
Re: What is an eigenvalue?
#183blue1brown on this:
Re: What is an eigenvalue?
#19Earlier 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?
Center of mass; the object itself.
Re: What is an eigenvalue?
#20I do think, the term eigenvalue is rather opaque, and should be replaced by a more plain-english terminology that readily conveys its meaning.
Try to name the bones you used to type this sentence?