What is an eigenvalue?
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What is an eigenvalue?
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Re: What is an eigenvalue?
#2Does 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.
Re: What is an eigenvalue?
#3Does 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.
Re: What is an eigenvalue?
#4Does 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.
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Re: What is an eigenvalue?
#5Does 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.
Most are introduced to the interplay between physics and linear algebra through the study of the mass - spring system where the type (real or complex), sign and amount of the eigenvalues determine its behavior and stability. For example, complex eigenvalues with positive real part indicate an unstable, or chaotic, in terms of amplitude convergence oscillation.
Re: What is an eigenvalue?
#6Does 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.
If i recall correctly you can represent a harmonic oscillator as a linear differential equation with a 2x2 matrix. The imaginary part of the eigenvalues of this matrix correspond to the angular frequency of the oscillator.
I like this example because it gives a physical meaning to both eigenvalues and imaginary numbers. It also shows the connection between the sine and cosine and the complex powers of e comes from (since you can show that all three solve the differential equation).
Re: What is an eigenvalue?
#7Does 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...
Re: What is an eigenvalue?
#8Does 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 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, that's an eigenvector (with eigenvalue 1), the vector perpendicular to the mirror is unchanged, but flipped, so that's an eigenvector (with eigenvalue -1).
If we dilate everything along the x axis by a factor of 2, say, then the x axis is an eigenvector (with eigenvalue 2), while the y and z axis and any vector in that plane is an eigenvector (with eigenvalue 1). Any other vector is "tilted", so not mapped to itself, so not an eigenvector.
Re: What is an eigenvalue?
#9Does 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.
When dealing with the Schrödinger equation, the eigenvalues are the energy levels of the quantum system.
Re: What is an eigenvalue?
#10Does 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 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?