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What is an eigenvalue?

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Re: What is an eigenvalue?

#31

Post by sfpotter »

_spduchamp wrote: post #13
qntty wrote: post #3

Earlier quoted context omitted.

Natural frequencies of mechanical systems are eigenvalues of it’s equation of motion.

Any references to help me unpack what you just said there?

Things that vibrate have natural modes of vibration. A particular vibrational pattern can be decomposed into a time-varying linear combination of these modes. The modes of vibration are eigenfunctions and the frequencies at which they vibrate are the square root of the corresponding eigenvalues.

You can look up a vibrating drum head (circular membrane) for a simple example.

Re: What is an eigenvalue?

#32

Post by cscheid »

bodhiandphysics wrote: post #29
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.

Also in computer science.. a web sites page rank is the eigenvalue of the connectivity matrix.

That's not true. The page rank is read from the eigenvector, and is the value associated with the given vertex (ie web page). There are as many page rank values as there are web pages, but only one eigenvector from which to read: the dominant eigenvector of the transition matrix, which is the one with the largest eigenvalue. So, only a single eigenvalue for the entire pagerank computation.

Re: What is an eigenvalue?

#34

Post by sfpotter »

_spduchamp wrote: post #12
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.

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.

See my other reply.

Re: What is an eigenvalue?

#36

Post by sampo »

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.

> Does anyone know of an example of a simple physical system where eigenvalues have a physical interpretation?

Oscillation modes in mass-spring systems. Here is a simple one with 2 masses and 3 springs, so the matrix is only 2-by-2.

https://math24.net/mass-spring-system.html

With more than 2 masses, you don't need to arrange the masses on a line, but you can have a 2d or 3d arrangement, with interconnecting springs. I am sorry I failed to find an example image.

The theory is explained, for example, around page 479 in this Thornton and Marion Classical Dynamics textbook. But you need to read about Lagrangian mechanics (chapter 7) before it makes sense.

https://eacpe.org/app/wp-content/uploads/2016/11/Classical-D...

Re: What is an eigenvalue?

#37

Post by syrrim »

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.

The vibration of a bell (say) could be modelled by a matrix, with a state vector to represent position, velocity, and acceleration, and the matrix modelling the differential equations describing their evolution over time. The eigenvectors represent a basis of the system, so that we can describe any potential state vector as a sum of eigenvectors. If we do so, then each step of the system can be modelled by multiplying each of these eigenvectors by its corresponding eigenvalue. If an eigenvalue happens to be complex, then we can describe it in phasor form as the product of an amplitude and a angle. The amplitude tells us how it will decay (or amplify) over time. The angle tells us the frequency of oscillation, and thus the note that the bell will typically sound.

Re: What is an eigenvalue?

#38

Post by DennisP »

MichaelZuo wrote: post #10
FabHK wrote: post #8

Earlier 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?

When you're rotating something, the axis of rotation. That's the point that doesn't change in rotation ("maps to itself").

Re: What is an eigenvalue?

#40

Post by gpsx »

billfruit wrote: post #16

I do think, the term eigenvalue is rather opaque, and should be replaced by a more plain-english terminology that readily conveys its meaning.

You know what's another one from physics whose name has nothing to do with the actual meaning - "Gedanken experiment"
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