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Earlier quoted context omitted.
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...
Earlier quoted context omitted.
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 th…
Does 'through the origin' imply motion through 'the origin'?
Who is this explainer aimed at? If you can understand the first sentence, you probably already know what an eigenvalue is.
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.
Get a sheet of rubber. Grab it in both hands and stretch it. Inspect your sheet and find a line on the sheet that you could draw in with a marker and when you stretched the sheet the line would grow and shrink, but would not change what it was pointing at (probably a line from one of your hands to the other, in this simple example) That is an eigenvector of your sheet stretching transformation. The eigenvalue is how…
Who is this explainer aimed at? If you can understand the first sentence, you probably already know what an eigenvalue is.
Who is this explainer aimed at? If you can understand the first sentence, you probably already know what an eigenvalue is.
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.
However, if you write down the matrix of spring constants for the system and solve for the eigenvalues and eigenvectors of this system you can do something special. If you compress or stretch the molecule along the direction of the one of the eigenvectors then let go, the molecule will continue to vibrate along that same direction. The motion will not spread out to all other degrees of freedom. It will also vibrate with a frequency given by the eigenvalue of that eigenvector.
Additionally, any complex vibration of the system can be broken down into a combination of these independent vibrational modes. This is a simple fact because the eigenvectors form an orthogonal basis for the space.
I was blown away in my Digital Signal Processing (DSP) class that eigen "values" exist for certain systems in the form of "waves". Basicaly you put in a wave made from multiple sine and/or cosine waves through some function f(x) and the output is STILL a wave, though its frequency, amplitude and phase might change. Technicaly if I remember correctly this applies to all complex exponentials, since those can be rewritt…
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.
https://drive.google.com/file/d/12SM0SAOvMq166gc8B1b81Y_S7HP...
The third page in particular shows a plot of "amplitude" versus "frequency" to show the "harmonic spectrum of a sawtooth wave". The "frequencies" correspond to the modes of vibration (i.e., sine waves of different frequency), which are the "eigenvectors" in this case. The "amplitudes" are the relative contribution of those vibrations to the overall sound, and these correspond to "eigenvalues".
The article is talking purely about constructing sounds via synthesis, so there's not necessarily a linear system associated with it, but there is a connection. Wave equations represented by linear partial differential equations can often be analyzed as a linear system that has these "modes of vibration" (i.e., series of orthogonal sinusoids at different frequencies). If you were to, for example, model a plucked string (like a guitar), you can model the solution as a weighted sum of eigenvectors (in this case, "modes of vibration" or sinusoids of different frequencies). The "weights" would be the eigenvalues, which determine the spectrum and ultimately the timbre of the sound produced.
That might seem more involved, because it's an infinite-dimensional linear system (i.e., the vectors are functions on a interval, rather than finite lists of numbers). It turns out, though, that the finite-dimensional discretization of an infinite-dimensional linear system (i.e., a partial-differential equation approximated by a finite-dimensional linear system) will sometimes have eigenvectors / eigenvalues that have similar features as the infinite-dimensional case. For example, there are certain finite-difference operators that can be written in matrix form whose eigenvectors will work out to be sampled sinusoids.
I'm not totally sure of the history, but I think a lot of the interest in eigenvectors / eigenvalues as a topic in matrix theory originated from this are (i.e., numerical solutions for partial-differential equations that were used to model physical systems).